A geometric approach to homology theory

Full text

London Mathematical Society Lecture Note Series. 18
A Geometric Approach
to Homology Theory
by S.BUONCRISTIANO,
C.P.ROURI(E,
and B.J.SANDERSON·
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE
LONDON . NEW YORK . MELBOURNE

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Contents
©
Cambridge University Press 1976
Library of Congress Catalogue Card Number: 75-22980
ISBN: 0521 209404
Printed in Great Britain
at the University Printing House, Cambridge
(Euan Phillips, University Printer)
I
II
III
IV
V
VI
VII
Introduction
Homotopy functors
Mock bundles
Coefficients
Geometric theories
Equivariant theories and operations
Sheaves
The geometry of CW complexes
Page
1
4
19
41
81
98
113
131

Introduction
The purpose of these notes is to give a geometrical treatment of
generalised homology and cohomology theories. The central idea is that
of a 'mock bundle', which is the geometric cocycle of a general cobordism
theory, and the main new result is that any homology theory is a general-
ised bordism theory. Thus every theory has both cycles and cocycles;
the cycles are manifolds, with a pattern of singularities depending on the
theory, and the cocycles are mock bundles with the same 'manifolds' as
fibres.
The geometric treatment, which we give in detail for the case of
pl
bordism and cobordism, has many good features. Mock bundles are
easy to set up and to see as a cohomology theory. Duality theorems are
transparent (thePoincare duality map is the identity on representatives).
Thorn isomorphism and the cohomology transfer are obvious geometrically
while cup product is just 'Whitney sum' on the bundle level and cap product
is the induced bundle glued up. Transversality is built into the theory -
the geometric interpretations of cup and cap products are extensions of
those familiar in classical homology. Coefficients have a beautiful geo-
metrical interpretation and the universal coefficient sequence is absorbed
into the more general 'killing' exact sequence. Equivariant cohomology
is easy to set up and operations are defined in a general setting. Finally
there is the new concept of a generalised cohomology with a sheaf of co-
efficients (which unfortunately does not have all the nicest properties).
The material is organised as follows. In Chapter I the transition
from functor on cell complexes to homotopy functor on polyhedra is
axiomatised, the mock bundles of Chapter II being the principal example.
In Chapter II, the simplest case of mock bundles, corresponding to
pl
cobordism, is treated, but the definitions and proofs all generalise to the
more complicated setting of later chapters. In Chapter
III
is the geo-
metric treatment of coefficients, where again only the simplest case,
1

pl bordism, is treated. A geometric proof of functoriality for coefficients
is given in this case. Chapter IV extends the previous work to a general-
ised bordism theory and includes the 'killing' process and a discussion of
functoriality for coefficients in general (similar results to Hilton's treat-
ment being obtained). In Chapter V we extend to the equivariant case and
discuss the Z
2
operations on pl cobordism in detail, linking with work
of tom Dieck and Quillen. Chapter VI discusses sheaves, which work
nicely in the cases when coefficients are functorial (for 'good' theories
or for 2-torsion free abelian groups) and finally in Chapter VII we prove
that a general theory is geometric. The principal result is that a theory
has cycles unique up to the equivalence generated by 'resolution of
singularities'. The result is proved by extending transversality to the
category of CW complexes, which can now be regarded as geometrical
objects as well as homotopy objects. Any CWspectrum can then be seen
as the Thorn spectrum of a suitable bordism theory. The intrinsic geo-
metry of CW complexes, which has strong connections with stratified
sets and the later work of Thorn, is touched on only lightly in these notes,
and we intend to develop these ideas further in a paper.
Each chapter is self-contained and carries its own references and
it is not necessary to read them in the given order. The main pattern of
dependence is illustrated below.
I
t
7~T
IV V VI
~
VII
The germs of many of the ideas contained in the present notes
come from ideas of Dennis Sullivan, who is himself a tireless cam-
paigner for the geometric approach in homology theory, and we would
like to dedicate this work to him.
2
NOTE ON INDEXING CONVENTIONS
Throughout this set of notes we will use the opposite of the usual
convention for indexing cohomology groups. This fits with our geometric
description of cocycles as mock bundles - the dimension of the class then
being the same as the fibre dimension of the bundle. It also means that
coboundaries reduce dimension (like boundaries), that both cup and cap
products add dimensions and that, for a generalised theory,
n
h (pt.) ~ hn(pt.). However the convention has the disadvantage that
ordinary cohomology appears only in negative dimensions. If the reader
wishes to convert our convention to the usual one he has merely to change
the sign of the index of all cohomology classes.
3

I·Homotopy functors
The main purpose of this chapter is to axiomatise the passage
from functors defined on
pl
cell complexes to homotopy functors defined
on polyhedra. Principal examples are simplicial homology and mock
bundles (see Chapter II).
Our main result, 3.2, states that the homotopy category is iso-
morphic to the category of fractions of
pl
cell complexes defined by
formally inverting expansions. Thus to define a homotopy functor, it is
only necessary to check that its value on an expansion is an isomorphism.
Analogous results for categories of simplicial complexes have been
proved by Siebenmann, [3].
In §4, similar results are proved for A-sets. This gives an
alternative approach to the homotopy theory of A-sets (compare [2]).
In §6 and §7, we axiomatise the construction of homotopy functors
and cohomologytheories. Here we are motivated by the coming applica-
tion to mock bundles in Chapter II, where the point of studying cell
complexes, rather than simplicial complexes, becomes plain as the Thom
isomorphism and duality theorems fall out.
The idea of using categories of fractions comes (to us) from [1]
where results, similar to those contained in §4 here, are proved.
Throughout the notes we use basic
pl
concepts; for definitions
and elementary results see [4, 6 or 8].
1. DEFINITIONS
Ball
complexes
Let K be a finite collection of
pl
balls in some Rn, and write
IKI = u
{a: a
EK l. Then K is a ball complex if
(1) IK
I
is the disjoint union of the interiors
a
of the
a
EK, and
(2)
a
EK implies the boundary
a
is a union of balls of K.
4
It then follows that
(3) if
a,
T EK, then
anT
is a union of balls of K.
Notice that we do not assume
anT
EK.
A subset L
C
K is a subcomplex if L is itself a ball complex,
and we write (K, L) for such a pair. If (K , L ) is another pair, and
o
0
K
C
K, L cLare subcomplexes, then there is the inclusion
o
0 ----
(K , L )
C
(K, L). An isomorphism f: (K, L)'" (K , L ) is a
pl
o
0 1 1
homeomorphism f: IKI •.• fK
I
suchthat fiLl = IL I, and
a
EK
1 1
implies
f(a)
EK. In the case where K and K are simplicial com-
1 1
plexes, there are simplicial maps f: (K, L)'" (K , L). The product
1 1 ----
K x L of ball complexes K, L is defined by K x L =
{axT
I
uEK, TEL}.
The categories Bi and Bs
Now define the category Bi to have for objects pairs (K, L) and
morphisms generated by isomorphisms and inclusions (i. e., a general
morphism is an isomorphism onto a subpair). The category Bs has the
same objects but the generating set for the morphisms is enlarged to
include simplicial maps between pairs of simplicial complexes.
Subdivisions
If L', L are ball complexes with each ball of L' contained in
some ball of L and
I
L' I = IL
I,
we say L' subdivides L, and write
L' <l L. The categories Bi and Bs enjoy a technical advantage over
categories of simplicial complexes; namely, if L
C
K and L'
<J
L,
then there is a complex L'uK= {a:uEl' or K-Ll.
Collapsing
We assume familiarity with the notion of collapsing, as in [6],
for example. Suppose (K , L )
C
(K, L), where L = L n K ; then we
o
0 0 0
have a collapse (K, L)' (K , L ) if K '" K and any elementary col-
---- 0 0 0
lapse in the sequence from a ball in L is across a ball in L (so that
in particular, L' L). We call the inclusion (K , L )
C
(K, L) an
o
0 0
expansion. The composition of an expansion with an isomorphism is still
called an expansion.
5

2. SUBDIVISIONIN THE CATEGORYOF FRACTIONS
Let B
=
Bi or Bs, and let 2.; denote the set of expansions. The
category of fractions B[L-
I]
is formed by formally inverting expansions.
Thus the objects are the same. New morphisms e
-1,
eEL, are intro-
duced, and a morphism in the category of fractions is then an equivalence
class of formal compositions g
0
go ...
0
g , where g. EB or
1 2
n
1
g.
=
e~
1
for some e. EL. The equivalence relation is generated by the
1 1 1
following operations:
(i) replace h by fog if h
=
fg and f, g EB;
-1 -1
(H) introduce e
0
e or e
0
e, eEL;
-1 -1-1
(iii) replace (e e) by e
0
e .
2 1 1 2
In fact operation (Hi) is a consequence of operations (i) and (H). Denote
the equivalence class of a formal composition by {g
0
go ...
0
g }.
a
1
n
The category of fractions is characterised by a universal map-
ping property; namely, given any functor F: B -+ C such that F(e)
is an isomorphism for each eEL, then there exists a unique functor F'
so that
B
commutes, where p is the natural map.
For simplicity, in the rest of the paper we will ignore pairs
(K,
L) with L"*
p'
when the general case can be obtained by making
minor adjustments. We first observe that any morphism in Bi[L-
I]
may
in fact be written {e
-1
0
f) by repeated use of the following lemma.
Lemma 2. 1. Let e :
J
-+K be an expansion and f :
J
-+L ~
mo!p~!sm i~ Bi. Th.~_~!1e_:r:~~_~?:p~xpa.~~!oneo and morphism fa so
that
6
J
;/~
K L
f~ A
commutes.
Proof. Define
J
=
K
U
L, where
J
is regarded as a sub-
a
J
complex of both by e and f. Then let e and f be the obvious in-
a a
elusions. That e is an expansion follows by echoing the collapse
a
K~J.
Remark. The lemma fails for Bs. For instance, take
K
=
I x I
J
=
I
O}
x I and L
=
I0}. Then f must be degenerate on
" a
eJ and hence be degenerate on the 2-cell in K.
Now suppose L'
<l
L; then there are inclusions i: L-+LXIuL'x
11}
and e : L' -+L x I
U
L' x {I}. Then e is an expansion, and so we have
a morphism Ie
-1
0
i) : L
-+
L' in B[L-
I],
called subdivi~lon and
denoted
[>
(L, L').
Lemma
2. 2.
C>
is function~~_!ha~~~!
(i)
C>
(L, L)
=
idE!~ti!y~
(H) if L"
<l
L'
<l
L, !!!en
[) (L', L")
0
[>
(L, L')
=
C>
(L, L").
Proof. For part (i), we must show that if i , i :K -+K x I
a
1
are the inclusions, then {i }
=
{i ). This is proved by simple col-
a
1
lapsing arguments. First, consider K x I subdivided to (K x I)' by
placing K at K x
H
1.
There are then inclusions i~, i~, i~ of K in
(K x I)' and a reflection
1':
(K x
1)'
-+ (K x I)' about the half-way level.
Now ri.!.
=
i.!., and {i.!.} is an isomorphism since (K x I)' '" K x {~ }.
222
It follows that II'}
=
identity. Since ri'
=
i
l,
we have Ii'}
=
{i'
1.
a a
1
The result then follows by considering K x
t::.
2•
This argument is
7

essentially due to Siebenmann [3; p. 480].
For part (ii), let
1:.2
be a 2-simplex with vertices v , v , v
012
. 111 2 2 1
and OpposItefaces
Il ,
I:. ,
1:..
Let (L x
Il )'
=
L x
I:.
U
L' x
Il U
o
1 2 0
L" x {v
1.
Then (L x
1:.2)'
'X
L'
X III U
L
X
1:.
1
U
L" x {v }~ L"x {v )
2 0 1 2 2
The proof is completed by a diagram chase.
See
Fig. 1.
L"
•
<'I; ~
(a
x
I)
u
a"
x {
1 } '"
(a
x
I) U
(J
x
{o }
U
a"
x
{1 } '"
(aX
I)
u
a
x {
0 },
c
where the first collapse is elementary, and the second is cylindrical.
Corollary 2.4. Suppose L'<l L; then
C>
(L, L') :L-L' is
an isomorphism.
Proof. It follows fr om 2. 3 that there is L" <l L' and L'" <l L",
so that
[>
(L, L") and
[>
(L', L'") are isomorphisms. The result now
follows from 2. 2.
~
L'·-
Fig. 1
t
3. ISOMORPInSMWITH THE HOMOTOPYCATEGORY
Nowlet Bh denote the category with objects pairs of ball com-
plexes, and morphisms homotopy classes of continuous maps. Then there
are natural maps
Bi
n -----
Bh
BS----
and by the universal mapping property, we have a diagram
Lemma 2. 3. Suppose given L' <l L; then there exists L" <l L'
such that (L x I)
U
L" x {l} '" L x
{O
l.
Proof. Let
aI' ... ,
an
be the balls of L listed in order of
decreasing dimensions. We subdivide L in n steps. After step r,
the interiors of
a , ... , a
are not touched again. Suppose r-steps
1
r
completed, and let
a=
ar
+
l• Then
a
has been subdivided to
a',
say.
Let
T
£
a'
be a cell with dim
T
=
dim
a,
and
T
n
a
=
¢.
If no such
T
exists, perform a preliminary subdivision of
a.
Nowassume there is
such a T. Then
a
=
a -
interior
(T)
is a collar on
a
in
a,
and
c
a
c
a',
Subdivide
(J
to
a"
so that a collar projection
a" -
a"
is
c c c c
simplicial.
a"
=
T U
a~
is the required subdivision of
(J.
Note the
cylindrical collapse
a" ~
a".
c
The resulting complex L" clearly has the desired property since
8
(3. 1)
Theorem 3.2. The maps in diagram
(3,1)
are isomorphisms of
categories.
Proof. Since all three categories have the same objects, it
suffices to show that each of
0,
f3,
y
is an isomorphism on the set of
morphisms from K to L for any K and L. We prove this in three
steps:
A.
0
is surjective,
B.
0
is injective,
C.
f3
is surjective.
9

The result then follows by commutativity. We first observe that
a
and
y
are compatible with subdivision; 1.e. ,
Remark 3.3. Suppose L'
<J
L; then
a(e>
(L, L')) is the homo-
topy class of the identity map.
~tep~
a
is surjective.
Suppose [f]: K ...•L is a homotopy class; then by the pl approxi-
mation theorem,
*
there exist subdivisions K' <l K, L' <l L, and a
simplicial Ii1ap f' : K' ...•L' so that [f'] = [fl. Let M(f') be the sim-
plicial mapping cylinder of f'. There is then an inclusion i: K' ...•M(f'),
and an expansion e : L' ...•M(f'). It follows from 3.3 that
f e
K~
0
,.
J
0
L
••
7
0
'
II
(K x I)'
c
Mf
:=J
J'--<J
L
A
!~
g
/I
K f
,.
J .••
L
1
e
1 1
Commutativity in Bi[~-I] follows from definitions and 2.2.
(Note that the left-hand half commutes by 2.2(0.) The map g is an iso-
morphism by 2. 2 and 2. 4.
The result follows.
[f]=a([)
(L, L'fl{e-
1Q
il[) (K, K')).
step B.
a
is injective.
By
3.1,
it is sufficient to show that for any diagram
Step C.
{3
is surjective.
It is sufficient to show that the class {f
1
of a simplicial map
f : K ...•L is in the image, since any map in Bs is a composition of maps
in Bi and simplicial maps. Consider the commutative diagram
4. A-SETS AND THE CATEGORY OF FRACTIONS
(K
x
1)' - M
f' f
K • L
f
We assume familiarity with the basic definitions involving A-sets
which are found in [2]. An inclusion of A-sets L
CK
is called an ele-
ment~y exp~nsion if K is obtained from L by attaching a set of sim-
Ue
K
~ inclusion
i
r;
o
U
i
1
where Mr is the simplicial mapping cylinder of f, and (K x 1)' is
K x I derived at K x {~l. The simplicial map f' is defined using the
obvious vertex map. Then i and e are expansions, and we have
-1 _10-1 -1-1
{f)={e of'oi l={e otoi oi
)={3{e
otoi oi
1,
1 0 1 0 1
since each map in the bracket lies in B1.
*
This is weaker than the usual simplicial approximation theorem. See
[4] or [8]for a short proof.
J
1
in which the e. are expansions, and such that
a
{e-
1
0
f
1
=
a {e~1
0
f
1
l.
_11 -1 -1 0 0_1
we have {e
0
f ) = {e
0
fl. Now
a
{e
0
f
1
=
a
{e
0
f
1
00 11 00 11
means that the diagram homotopy commutes, provided we regard eo
and e as homotopy equivalences.
1
Nowlet
J
=
J
uL
J.
Then by homotopy commutativity and the
o
1
relative pl approximation theorem (see footnote ), there is a sim-
plicial map f :
(K
x I)' ...•
J'
so that f
I
/KI
x Ji
1
= f., i =
0,
1. Now
1
consider the following diagram in which arrows marked e are expansions,
and arrows marked
[)-+
are compositions of subdivision followed by
inclusion:
J
yO~
K L
~~1
10
11

Proof. We first show that l/I is surjective. An element of
{X, K} has a representative e-I
0
f, where f is a Li.-mapand e is
an expansion. Using the Kan condition on K, we find a Li.-map r so that
roe
=
id. Define g
=
r
0
f. Then {g)
=
{r
0
f)
=
{r
0
e
0
e- 1
0
f )
=
{e-
I
0
fl.
plexes ~n(s), for s in some indexing set, to L via Li.-maps
f : A
,is) -
L. An inclusion L
C
K is an expansion if there is a
s
11,1 -- ...
n' _
countable sequence L c. L
C
L
C
L ... of elementary expansions so
123
that K
=
U
L,. In particular, the 'end inclusions' of K in K
@
I are
1
easily seen to be expansions. Let L: denote the set* of Li.-maps which
are expansions, and let ~[L.;-I] denote the resulting category of fractions.
The analogue of
2.1
is easily proved for Li.-sets so that any morphism in
~[2:;-I] may be written {e-1ofl. Let ~h denote the category of Li.-sets
and homotopy classes of continuous maps between realisations.
X
---- .•,- K
Theorem 4.
1.
The canonical functor Li.(L:-1)- Li.h is an iso-
--
Proof. The proof is analogous to the proof of 3.2. The main
modifications are to replace the cell subdivisions used hy simplicial ones.
This can be done by deriving. Also, the Li.-sets need to be replaced by
simplicial complexes before applying the simplicial approximation
theorem,
t
This is done by using (L x
1)'
with L x
{1)
derived twice,
and L x {~} derived once. We leave the reader to check details.
Nowlet K be a Kan Li.-set, and let [X, K] denote Li.-homotopy
classes of Li.-maps X- K; let {X, K} denote the set of morphisms
X- K in ~(L:-I). From Theorem 4.
1,
we have a well-defined function
l/I: [X, Kl- {X, K). In fact, l/I can easily be well-defined without the aid
of 4.
1,
and the proof of the following theorem is then independent of 4.
1.
g
Thus l/I[g]
=
{e-I
0
.f
l.
To see that
lJI
is injective, suppose
that lJI[go]
=
lJI[g) This means that gl is obtained from go by a
sequence of the following steps and their inverses:
(i) replace (fg) by fog,
(ii) introduce e-I
0
e or e
0
e-I
,
(iii) replace (e e fl by e-I
0
e-I
2 1 1 2'
Now it is easy to see that the map g defined above is unique up
to Li.-homotopy; use the expansion L x {O)
U
K
@
I
U
L x
{1)
-L
@1.
Hence after each step (ii), we can map each A-set in the composition into
K uniquely up to Li.-homotopy. It follows that g and g are the same
o
1
up to Li.-homotopy.
Now let CWh denote the category of CW complexes and homotopy
classes of maps. Then there is the composition
-I
f :
~[L: ] -
~h - CWh.
Theorem 4.2. Suppose X is a Li.-set and K is a Kan Li.-set.
Then the canonical function Theorem 4. 3.
-I
f :
~[L: ] -
CWh is an equivalence of categories.
l/I : [X, K] ....• {X, K}
is a bijection.
* P. May has pointed out that there are set theoretic problems here.
We adopt MacLane's axiom of one universe [7; p. 22]. Theorem 4.
1
then shows that ~[L:-I] is a category in the universe.
t
The required approximation theorem is given in [2; 5.
1].
12
Proof. It is sufficient to show that if X is a CW complex, then
the natural map
I
S(X)
I ....•
X is a homotopy equivalence. From definitions,
we have an isomorphism
fJ
0
n:
a
f3
1T
SX -
1T
Isxl -
1T
X.
n n n
But
a
is an isomorphism by a relative version of 4. 2. It follows that
f3
is an isomorphism, and the result follows from
J.
H. C. Whitehead's
13

theorem [5J.
Remark 4.4. There are easily provable relative versions of the
theorems presented in this section.
5. HOMOTOPYFUNCTORS
We rephrase Theorem 3. 2 in terms of functors defined on
B
=
Bi or Bs. Let T : B - C be a functor.
Ax C(C 11 ) L t e (K L) ....•(K L) be an expansion. Then
_lOm __
0
apse. e :
0' 0 '
T(e) is an isomorphism.
From the universal property of the category of fractions and
Theorem 3.2, we have
Theorem 5.
1.
Any functor T satisfying C factors uniquely
through Bh; moreover, if T is defined on Bi, then it extends uniquely
to Bs.
Suppose now T is contravariant and satisfies Axiom C. Denote
the canonical extension by T also. Suppose
I
L
I
=
IK
I;
then we have
an isomorphism
T[id] : T(L) - T(K).
If L
<J
K, we call this an amalgamation isomorphism and write
am :
T(L) - T(K).
From definitions, we see that am
=
T(I>(K, L)). The inverse isomor-
phism is called a subdivision isomorphism and we write
sd : T(K) - T(L).
Nowlet Ph be the category of compact polyhedra and homotopy
classes of continuous maps. Let T : Bi ..• S* be a functor satisfying
Axiom C, where S* is the category of based sets. We define T: Ph -S*
as follows.
An element of T(P) is an equivalence class of elements of T(K),
where IKI
=
P. Suppose IKil
=
P, i
=
0, 1. Then
U
o
E
T(K
o)
is
14
defined to be equivalent to u
E
T(K) if there exists u
E
T(K') for
1 1
some K'
<J
K., such that am(u.)
=
u, i
=
0, 1. Alternatively,
1 1
T(P)
=
lim T(K) where the limit is taken over all K with IKI
=
P.
-
The canonical function T(K) - T(P) is of course a bijection. If
[f) : P -
Q
is a homotopy class, then T[f] is defined by choosing a
representative for [f) in Bs.
Finally, note that for an arbitrary category C, we could define
T : Ph - C by choosing for each P a particular K with
I
K
I
=
P, and
then defining T(P)
=
T(K). Any two such choices give naturally equiva-
lent functors.
6. CONSTRUCTIONOF FUNCTORS
Suppose given a contravariant functor Z : Bi - S* satisfying the
following axioms.
E (extension). Suppose e : (K , L ) - (K, L) is an expansion; then
o
0
Z(e) : Z(K, L) ....•Z(K , L ) is surjective.
o
0
G (glue). Suppose K
=
K UK, L
C
K, and L.
=
L n K., i
=
1, 2.
1 2 1 1
Suppose u.
E
Z(K., L.), i
=
1, 2, restricts to u
E
Z(K
fI
K , L ), where
1 1 1 1 2 0
L
=
L n (K n K). Then there exists z
E
Z(K, L) so that z restricts
o
1 2
to u., i
=
1, 2. Moreover if K n K
C
L then z is unique.
1 1 2
Define T(K, L)
=
Z(K, L)/-, where z _ z if there is a
o
1
z
E
Z(K x I, L x I) so that z. is identified with the restriction
1
z
I
E
Z(K xli}, L x Ii} ~ Z(K, L). It is an easy exercise to show
that - is an equivalence relation. To see that z - z, consider
Z(K x ~ , Lx ~ ).
2 2
Now if f : (K , L ) ....•(K , L ) is a morphism in Bi, then
1 1 2 2
T(f) : T(K , L ) - T(K , L ) is clearly well-defined by
2 2 1 1
T(f)[z]
=
[Z(f)z].
Proposition 6. 1. The functor T satisfies Axiom C.
Proof. Let e : (K , L ) - (K, L) be an expansion. We have to
o
0
15

show T(e) is an isomorphism. But by Axiom E we have T(e) is onto.
Suppose then that T(e)z
=
T(e)z. Construct z
E:
Z(K x I, L x I) such
o
I
that z : z _ z in two steps. First, find z
E:
Z(K U(K XI), L U(L xI»
o
I 2 0
a
using Axiom G twice; then find z using E.
Lemma 7. 1. T satisfies the following axioms for any
K=>K,K:
I 2
Half exact: T(K, Kl UK)"'" T(K, KI)"" T(K
2,
Kl n K)
is exact.
Compatibility of extension to Bs
Nowsuppose that Z is in fact defined on Bs. Then we have T
defined using Z
I
Bi, and T extends uniquely to Bs by Theorem 5. 1.
Proposition 6. 2. The extension of T to Bs is given by
T(f)[z]
=
[Z(f)z].
Proof. By uniqueness, it suffices to show that T(f) is well-
defined on Bs by the above formula, and it is sufficient to consider the
case f simplicial. Let K be any cell complex, and define (K x I)'
by deriving each cell on the half-way level. Then if K is simplicial,
so is (Kx 1)'; and if f: KI - K
2
is simplicial, then the deriveds may
be chosen so that L(f x id) : (K x I)' ....•(K x I)' is simplicial. The
I 2
result therefore follows from
Excision: T(K UK, K ) ...•T(K , K n K )
I
2
I
2
I
2
is an isomorphism.
Proof. Order 2 is obvious; to see exactness, use E to extend
a concordance on K
2
to one on K. For excision, use the definition of
Z(K, L) and G.
Now suppose given functors zq for q
E
Z' defined on Bi and
extended to Bi as above, and suppose that in addition we have
Axiom S (suspension). * There are given natural isomorphisms
Then we can define
7. COHOMOLOGYTHEORIES
Now let Bi
C
Bi be the subcategory consisting of pairs (K, L)
with L
=
¢,
and suppose that Z : Bi - S* is a functor. Then we can
extend Z to Bi by defining Z(K, L)
=
Ker {Z(K)- Z(L)
l.
Suppose Z
now satisfies axioms E and G. Let T denote the associated homotopy
functor.
Proof. Consider
Q
=
(K X
~2),
obtained by deriving each cell
of K x {v }. Then Qcontains isomorphic copies of (K x 1)'; namely,
I
K x ~l and (K x (~l U~l»,. Moreover, it is easy to see that Qcol-
I 0 2
lapses to both these subsets. Therefore, given a z
E:
Z(K x I), we get
"i
E:
Z(K
X ~2)"
and hence
"il
E
Z(K x I)' and vice versa.
{o} )
T(i)
Lxi)
+-
Tq-1(W, KULX
1
TO)
T
q-
1(K, L)
to be the composition
sq
Tq(L) ...• Tq-1(L x I,
* See the note on indexing cohomology groups at the end of the introduction.
where W
=
K ULx {1}L x I, and i is an excision,
j
extends the identi-
fication L - L x
{O},
and as a map K"'" W is homotopic to the inclusion
by an extension of the obvious homotopy on L.
Then easy arguments show that the long sequence is exact, and
we have shown
if and only if there is z
E:
Z(K x I)' such
z - z
o
I
Lemma 6.3.
that z.
=
Z
I
K x {i
l.
1
16 17

Theorem
7.2.
A sequence of functors zq: Bi
-+
S*, q
E
Z,
sat~fying E, G, and S defines a cohomology theory on the category
~~~ll1pact polyhe~al pair~
Remarks
7.3. 1.
Tq() is in fact an abelian group functor.
This is seen by 'track addition': Given ~,
1/ E
Zq(K), form
sq~, sq."
E
Zq-I(K x I, K x
i),
and use G to construct
sq~
+
sqTj
E
Zq-I(K x I)'. Finally, use amalgamation and inverse of
suspension to return to Zq(K).
2,
In fact, half exactness, excision, and suspension imply co-
homology theory by formal argument, using Puppe sequences. Thus
Axiom S need hold only for Tq(,).
3,
A classifying &I-spectrum for Tq() can be constructed by
taking ail-set g with gk
=
zq(~k) then it can be seen that
I
g
I ""
n
I
g
11.
q We will ~XPlain this construction in detail in Chapter
q q-
t
'f"
't
II §5 for a specific example. This extends the theory
0
In Inl e com-
plexes.
REFERENCES FOR CHAPTER I
[I] P. Gabriel and M. Zisman. Homotopy theory and calculus of
fractions. Springer-Verlag, Berlin
(1967).
[2] C. P. Rourke and B.
J.
Sanderson. ~-sets
1.
Quart.
J.
Math.
Oxford Ser-,
2 (1971), 321-8.
p]
L. Siebenmann. Infinite simple homotopy types. Proc. Kon. Ned.
Akad. (Amsterda!!!1
73 (1970), 479-95.
[4]
J.
Stallings. Tata Institute notes on polyhedral topology,
1969.
[51
J.
H. C. Whitehead. Combinatorial homotopy
1.
Bull. Amer.
Math. Soc.
55 (1949), 213-45.
[61
E. C. Zeeman. Seminar on combinatorial topology. I. H. E. S.
and Warwick notes
(1963-69).
[71
S. MacLane. Categories for the working mathematician.
Springer- Verlag, New York
(1971).
[8J C. P. Rourke and B.
J.
Sanderson. Introduction to piecewise-
linear topology. Springer-Verlag, Berlin
(1972).
18
II-Mock bundles
We describe here the theory of mock bundles. This is a bundle
theory giving rise to a cohomology theory which in the simplest case is
pi cobordism. In this interpretation, all the usual products have simple
definitions, and the Thorn isomorphism and duality theorems have short
transparent proofs. Another feature is that mock bundles can be com-
posed yielding a COhomology transfer. The theory also provides a short
proof of the pi transversality theorem
[12; 1. 2].
In a final section,
classifying ~-sets are constructed. The construction is similar to
Quinn's [7; §1].
1. MOCK BUNDLES AS A COHOMOLOGY THEORY
Let K be a ball complex. A q-mock bundle* ~q/K with base
K and total space E ~ consists of a pi projection p ~ : E ~
-+
I
K
I
such
that, for each
a
E
K, P~
1
(a)
is a compact pi manifold of dimension
-1 -1
q
+
dim
a,
with boundary p ~
(u).
We denote p ~
(a)
by
~(a),
and call
it the block over
a.
The empty set is regarded as a manifold of any dimension; thus
Ha)
may be empty for some cells
a
E
K. Therefore, q could be nega-
tive, and then
~(a)
=
¢ if dim
a<
-q. The empty bundle ¢/K has the
empty set for total space, and is a q-mock bundle for all q
E
Z.
Figure 2 shows a I-mock bundle over the union of two I-simplexes.
Mock bundles ~, 17/K are isomorphic, written ~ ~
1/,
if there
is a pi homeomorphism h: E ~
-+
E
1/
which respects blocks; i. e. ,
h(~(a))
=
1/(a)
for each
aE
K.
Now define the based set Zq(K) to be the set of isomorphism
classes of q-mock bundles over K with base point the empty bundle.
* The terminology 'mock bundle' is due to M. M. Cohen.
19

by E(~) = E(~ ) uhE(~ ), and define the projection of ~ inductively,
1 2
using the fact that cells are contractible.
For Axiom S (suspension), suppose given
UK.
Now form
s~1K x I by E(s~) = E(~) and Ps~ = p ~ x
{i
J;
i. e., place ~ over
K x I at the half-way level. Then s ~ is empty over K x
i
(and over
L x I if ~ empty over L), so we have a suspension map
Lemma 1. 2. Suppose IKI is a
pl
n-manifold, and
~IK
is a
q-mock bundle. Then E~ is an (n+q)-manifold with boundary p~
1
(a
I
K
I).
The inverse map is defined by composition with the projection
11 :
K x I -+ K; i.
e.,
given
1]
IK
x I empty over K x
i,
define
-1 -1 -1
S 1]
IK
by E(s 1]) = E(1]) and p(s 1]) =7T
0
P1]' It is trivial to check
that s
-1
is an inverse for s.
Finally, we have to check Axiom E (extension). In other words,
if K XK and ~
IK
is given, we have to construct
~IK
so that
o
0 0
~ ~ ~ I
K. By induction on the length of the collapse, it suffices to prove
o
0
the case when the collapse is elementary across a cell
0-
from a free
face
T.
Let J be the subcomplex
a -
T.
Then IJ
I
is a ball, and we
canidentify
(0-,
IJI) with (IJI XI, IJI X
{OJ).
Wethendefine
EW
= E(~ ) UE(~ IJ) x I identified over E(~ /J) = E(~ IJ) x
{O
J,
o
0 0 0
and let PI: = p(~ ) on E(~ ) and p(~ ) x id on E(~ IJ) x I. That ~
,,0 0 0 0
is a mock bundle follows from Lemma 1. 2 below.
Fig. 2
Proof. (Compare [4;p. 142].) Let x EE(~); then xEint~(o-),
say. We can then define a 'transverse star' to x in E(~) by inductively
restricting collars of
a~(T)
in
~(T)
for
0-
<
T.
Then a neighbouthood
of x in E(~) is homeomorphic to {neighbourhood in ~(o-)
J
x
{transverse star
J.
The same construction holds for p(x) in K, and
the two transverse stars are abstractly isomorphic, hence homeomorphic.
But IKI being a manifold implies that the transverse star is a disc
Proof of 1 1 For Axiom G (glue), suppose given ~
l
1K
1'
~21K2'
(P
Q
0
if ld t ( ) if d 1 if pOt d
Q
•• X
lS a man
0
a x, y an on y lS one a x an one
d· hOh·I:IKnK~I:IKnK Form~IK=KUK
an an lsomorp lsm .
"1 1 2 "2 1 2' 1 2
at y; see [6]). The result follows on observing that x Eint (transverse
* See the note on indexing cohomology groups at the end of the introductiOi t ) if d 1 of
°
tiKI
saran on y
1 X
Em.
Theorem 1. 1. zq(,) satisfies axioms E, G, and S of Part I,
and hence, by I. 7. 2, {Tq(,), 0q
J
is a cohomology theory* on the cate-
gory of pairs of compact polyhedra.
case we now prove:
qht Blo of ball com-
Z () becomes a contravariant functor on t e ca egory
plexes and inclusions via the restriction: Suppose given
~IK
and L
C
K;
then
~IL
is defined by E(~IL)=p~l(L), and P(~IL)=p~I:E(~IL)-+L.
(We use both notations E(~) and E ~ for total spaces, etc., as con-
venient. )
Nowdefine a functor zq(,) on the category Bi of pairs of ball
complexes (as in I. 7) by defining Zq(K, L) to be the kernel of
Zq(K) .••Zq(L). In other words, Zq(K, L) is isomorphism classes of
bundles which are empty over L.
We can now define a functor Tq(,), as in I. 6, by taking Tq(K, L)
to be the set of cobordism classes of mock bundles empty over L, where
I: is cobordant to ~ written ~ _ ~ if there is a mock bundle
"0 l' 0 l'
1]1K
x I, empty over L x I, such that
1]
IK x li
J ~
~i for i = 0, 1.
It is easy to see that cobordism is an equivalence relation, and in any
20
21

The Thorn isomorphism
We
finish the section by observing that the proof of Axiom S given
above generalises at once to give a Thorn isomorphism theorem for pl
block bundles (see
[11]
for definitions).
Let ur
,1(
be a block bundle; then we can give E(u) a ball com-
plex structure in which the blocks of u are balls, for we merely have to
choose a suitable ball complex structure on E(u). Then we can define
Amalgamation
Let ~q
IK'
be given where K'
<l
K. Then p ~ : E (~) .••
I
K
I
is
a mock bundle over K by Lemma 1. 2, called the amalgamation of ~
and written am( ~). To see compatibility with amalgamation as in
Chapter I §5, notice that the bundle 1]
IK
x I
U
K' x
{O}
obtained by
extending
~IK'
x
{D)
via the proof of
1. 1
has total space homeomorphic
with E(~) x
1.
This is checked by induction over the skeleta. Uniqueness
of collars is used to match product structures.
Subdivision
Theorem 2.
1.
Suppose given
~IK
and K'
<l
K. Then there
exists
~'IK'
such that am( ~,) ~ ~.
Proof. An inverse for <J>is given by composition with a pro-
Addition
2. THE GEOMETRY OF MOCK BUNDLES
Nowlet
UK
and K'
<l
K. Examine the proof of existence of a
subdivision
~'IK'
such that am(~') - ~ (which follows from Chapter I
and
1. 1).
The proof says consider K"
<l
K' such that K", as a sub-
division of K, is cylindrical in each ball less a smaller ball (see
1.
2. 3).
Then K" x
{D)
u K x I 'K x {I} so that we can extend
~IK
x {I}
to
TJ
IK"
x
{D)
u K x I and let ~'= am(1]
I
K" x
{D)).
But the extension
is defined skeletally, and, on a typical cell,
a
E
K is obtained by first
extending over the cylindrical collapse
an - a ,
a
and then over the
1
initial collapse
a
x I '"
a" - a
u
a
x I u
a
x {I}.
1
It follows that we can inductively identify E(1]) with E(~) x I
since E(1]
I
a" - a ) ~
E(TJ
I
a)
x I can be identified with a collar on
~(a),
. 1
and 1]
(a
x
1)
=
Ha)
XI (by the proof of
1.1).
A simple collaring argu-
We now give geometric interpretations to parts of the cohomology ment is again used to match product structures. We have proved:
theory Tq(,) defined in
§1.
jection for u.
Remark. In the next section, we show that the Thorn isomor-
phism is given by cup product with a Thorn class.
Proposition
1.
3. <I>is an isomorphism, an~J:nduces an isomor-
q q-r .
phi~m~_called the ThOIl1_~omorphism, T (K)'" T (E(u), E(u)).
to be composition with the zero section i: K'" E(u); i. e., E(<l>(~))=
EW
and p(<I>(~))= i
0
p~.
E(f#(~)) - - - ~ E(~)
~ f
~p~
K ------. L
Suppose f: K .••L is a simplicial map between simplicial com-
plexes, and
~;l,
a mock bundle. Then we can form the pull-back diagram
t?:
t.
P1](x, t)=(P~(x), t),
(p
~(x),
1-
t),
There is an addition in Zq(K, L) given by disjoint union; i. e. ,
E
U
+
7])
= E(~)
U
E(1]) and p
+
= pup . It is then easy to check that Remark. The reader can check that this proof of existence of
this coincides with the 'track ;dd7tion' ~defin:d in Tq(,)
(1. 7.
3). To subdivisions is essentially that given in [9; p.
128].
see the group structure directly, observe that ~
+ ~-
¢
by letting Induced mock bundles
1J1K
x I have E(1]) = E(~ x I) and
22
23

and it is not hard to show that f
#
(s) is a mock bundle, and that
is well-defined and functorial (compare
[12; 2.3]).
This means that
zq( ,) and, hence by Chapter I, Tq( ,) is in fact defined on the larger
category Bs, and it follows from 1.6.
2
that the notion of pull-back is
compatible with the pull-back as homotopy functor.
The external product
Let sq
IK
and 1)r;1.. be mock bundles. Then s x 1)
IK
x L is
defined by E(~ x 1)
=
EW
x E(1) and p(~ x 1) ~ p ~x P1)' The blocks
of ~ x 1) are then products of blocks of ~ and 1). We thus get an
external product
Now if we subdivide 1) so that the blocks of ~ are subcomplexes, then
corresponding to
a
E
K, we have by 1.
2
the manifold E(1)
I
s(a».
In
other words, we have a (q+r)-mock bundle
which we call s
0
1).
P
E(1)
1).
EW
p~!p~
K
This extends to give a transfer
Remark. In the case that 1) is the identity id: L
-+
L, we have
[s x
1/]
=
11*[~]where 11:K x L
-+
K is the projection. To see this,
suppose K, L, and 11: (K XL)'
-+
K are all simplicial. Then 11#(~)
is a subdivision of ~ x 1).
where
where
p
=
Ps'
We observe that the transfer is functorial i. e.
p and q are mock bundle projections
The internal product
Suppose given ~q
IK
and 1)r/K. Define [~]
U
[1)]
E
Tq+r(K) to
bi
.;).*[~
X
1)], where .;).: K
-+
K x K is the diagonal map; i. e., subdivide
~ x 1) so that .;).(K)is a subcomplex, and then restrict to .;).(K).
The internal or cup product makes T*(K) into a commutative
ring with unit. To see that the class of id: K ..• K is the unit, use the
remark on external products and the fact that
11
0 .;).
=
id. Associativity
and commutativity are easily checked. There are natural relative ver-
sions of both products which we leave the reader to formulate.
The composition and the transfer
We now generalise the composition used in §1 to give suspension
and Thorn isomorphisms.
Let ~q
IK
be a mock bundle (a block bundle with closed manifold
as fibre is a special case), and let 1)r IE(~) be another mock bundle.
24
The transfer can be seen to be the composition of the following:
Tr(E(m Thorn iso,;. Tr-s(E(V), E(V»2:.Tr-s(KXlq+s, K>4q+1)~~sP'Tr+q(K)
where E(s) is embedded in K x int(lq+s) with normal bundle
vs.
Alternative description of the cup product
Proposition
2.2.
Let' sq, 1)r
IK
be mock bundles. Then
m
U[1)]
=
P, p*[ s], where p
=
p .
• -- 1)
In other words: pull s back over E(1) and then compose.
Proof. We may suppose that K is simplicial. Subdivide p so
that p : E(1)'
-+
K' is simplicial, and subdivide ~ to s'
IK'.
Let
(Kx K)' be the subdivision of K x K given by
[14;
Lemma
1]
so that
~(K) is now a subcomplex. Consider the following diagram.
25

Thorn and the Euler classes are natural and multiplicative. We leave
the reader to check these facts.
Properties of the cup product and the transfer
We now give some properties which will be used in Chapter V.
Proposition 2. 4. Suppose given a pull. back i!iagr~!!!..
Proof. Consider the diagram:
with f simpH~~al.__Then f*p!
=
PI
f'*.
Proposition 2. 6. Let p be..!heprojection of a mock bundle 1].
Then p! (p*~ u
1;)
= ~
LJ
p! ~.
f'
-----.,- E(~)
____ f ._
1
P
L
'1'_
)0
E(p*~)
••
E(~)
~-
-
-i--
Jp~
--
E(I;) E(1])
.•...
K
)0
•
PI;
P
27
Proof. Subdivide so that p : E(~)' ...•K' is simplicial, then sub-
divide so that f : L' ...•K' is simplicial. Now E(f#~) is a linear cell
complex in a natural way and we may subdivide without introducing new
vertices so that p', f' are also simplicial. The result now follows from
definitions.
Remark 2. 5. In the case
I
K
I
=
M is a manti old, f is the
projection of a mock bundle 1];K+, where K
<1
K+, and f, pare em-
#
beddings, then E(f ~) is the transverse intersection of
EW
with E(1])
'in M. The proposition then implies that
if
g: W ...•E(~) is a map, then
we can make g transverse to E(f#~) in E(~), or make g transverse
to E(1]) in M. The result in either case is the same. We return to
these ideas in §4.
Proof. The result follows from the alternative definition of the
cup product, and the fact that the Thom isomorphism is composition with
t(u). See diagram.
26
Proposition 2. 3. u t(u) : Tq(E(u)) ...•Tq-r(E(u), E(u)) is the
restriction isomorphism i* : Tq(E(u)) ...•Tq(K) composed with the Thorn
~~om.orphism.
E(i*(~)) ---~~~ E(~)
1 ~))
!
K ~ E(u)
Let ur;K be a block bundle with i: K ...•E(u) the zero section.
If E(u) is given the ball complex structure of §l (in which blocks are
balls), then i is the projection of a mock bundle, and thus determines a
class t(u)
E
T-r(E(u), E(u)), called the Thom class of u.
There is also the Euler class of u, e(u)
=
i*t(u) which can be
thought of as the result of intersecting K with itself in E(u). Both the
The squares are pull-backs, and K" is constructed as follows. Choose
a triangulation of K x E(1]) so that the blocks of id x
P'1/
are sub-
complexes. Then choose a further subdivision, and choose .K"
<1
K' so
that 1r is now simplicial. Then choose ~";K" subdividing ~';K'.
1
Without loss of generality, we can assume p~,
=
p~". Now P~ x P'1/ is
the projection of a mock bundle U(K x K)', since by construction
I; =
1r#(1])
0
1r#(~"). Butfrom definitions
P,
p*[~]
=
A*[
1;]=
A*[~x1]
]=[
~]u[~
2 1 .
The Thom class and the Euler class of a block bundle

Triangulate E(~), E(1/), K so blocks are subcomplexes and p~, pare
simplicial. Assume p~
=
p~
I
where K'
<l
K is the resulting subdivisiOL
The result now follows from definitions and 2. 2.
3. CAP PRODUCTS AND DUALITY
Let X be a topological space, and define
l'
(X) to be the set of
n
bordism classes of pairs (M, fl. Here M is a closed
pl
n-manifold,
f : M ..•X a continuous map, and (M , f ) is bordant to (M , f ) if
n+
1
0 0 I I
there exists f: W ..•X, where W is a
pl
manifold with boundary
the disjoint union W u W , and there are homeomorphisms g.: M. -W.
OIl 1 1
such that fog.
=
f., i
=
0,
1.
1 1
T (X) becomes a group by 'disjoint union' and is the nth
pl
n
bordism group of X.
There are relative groups
l'
(X, A) defined by considering maps
n
f : (M, aM)- (X, A) with the notion of bordism enlarged to allow
aw
=
w u W
U
W when Wand Ware disjoint, and f(W )
C
A.
o
I 2 0 I 2
There is a homomorphism a :
l'
(X, A)'" T l(A) given by
n n n-
restriction. The following theorem is well-known. A sketch proof is
included, designed to generalise in Chapters ill, IV.
f
=
f 1M defines an element of
l'
(X - U, A - U), and
I I
n
f
0
7T :
M x I
=
W..•X defines a bordism between f and f where
I 1
W
=
M x
{o
J,
W
=
M x {I
J.
To construct M , give M a metric,
o
I I I
and let e:
=
d(M - int(A },
cl
(U
n,
and triangulate M with mesh
I I
<
e:j2. Define P
=
u of closed simplexes which meet M - A , and
I
let M
=
2nd derived neighbourhood of P (cf.
[14]).
The required
1
properties are then easily checked. Injectivity follows by a similar
argument applied to a bordism.
The cap product
Let f : Mn..• K be a map, where M is a closed pl n-manifold,
and let ~q/K be a mock bundle. Form f*[ ~]/M, and choose a repre-
sentative 1/. Then by
1.
2, E(1/} is a closed (n+q}-manifold, and there
is a map g: E(1/}'"
I
K
I
by composing. Notice that if f is simplicial,
#
then we can take 1/
=
f (~).
Theorem 3.
1.
of topological pairs.
{T ( , ); a
J
is a homology theory on the category
n n .
We define [f] n [~]
=
[g], and the reader may check that we have
a well-defined cap product
Proof. Given a map f : (X, A)'" (Y, B), we get
f* : T (X, A) ..• T (Y, B) by composition. Naturality of f* and a are
n n
then obvious. Exactness is proved by easy geometric arguments. Hom-
otopy follows from the fact that M x I is an (n+l)-manifold with boundary
M x
{o
J
U
M x {I
J
u aM x I. It remains to prove excision. Let
U
C
A with
cl
(U)
C
int(A). Then we will show that
i* :
T (X - U, A - U) ..• l' (X, A)
n n
is an isomorphism.
To see surjectivity, consider f: (M, aM}'" (X, A), and define
U
I
=
f-
I
(U), A
=
C
I
(A). Then we have
cl
(U )
C
intM(A). We claim
I I I
that there is a manifold M
C
M, with M - A
C
M
C
M - U. Then
I I I I
28
Remarks.
1.
There are obvious relative versions of the cap
products.
2. The similarity of the cup and cap products (using the second
description of the cup product) is clear. This will become more trans-
parent later when they are seen to be dual.
Slant products
1.
Tq(X}
®
TS(X x
y} ..•
Tq+s(y}
[f]
® [~] •.•
7T
0
(f
x
ly)*[ ~].
29

In other words, take ~ x ly as a bundle over X x Y, take its
cap product with f, and then compose into Y.
Remark. As before, there are relative versions which give in
particular slant products when X is replaced by (X, x
o)'
Y by (Y,
Yo)'
and X x
Y
by X
A
Y'"
(X x
Y,
X
v
Y).
Poincare duality
f
x
ly
Consider M x
Y •
X x
Y
-+
Y.
Then we have the bundle
,~ ~.K
11
(f
x ly)"~;M x Y, which we can regard as a bundle over Y by com-
posing with
11,
Let M be a closed n-manifold and ~;M a q-mock bundle over
some complex underlying M. Nowthe fundamental (bordism) class [M]
of M is just the identity map
1:
M
-+
M, and since
1"
=
1,
[M] n ~ can
be interpreted as the same map p ~ : E(~) ....•M, but regarded as a bordism
class by
1.
2. In other words, we have the Poincare duality map
that f: W
-+
M is a map where W is an (n+q+l)-manifold with boundary
the disjoint union
EW
U E(1)).
We have to show that ~ -
1)
in Tq(K).
Using collars on the boundaries of W, we can replace f by a map
f :W ....•MxI sothat f-l(MXO)=~ and f-
1
(MXl)=1). Subdivide
1 1 1
f
1
so that it is simplicial, and so that the blocks of ~ and
1)
and the
cells of K x 0
U
K x
1
are all subcomplexes. Now consider the mock
bundle f : E(~) -
J
where
J
is the cell complex which has for cells
1
the duals in M x I to simplexes a
E
M x I and the duals in each cell
a
E
K x
(1
U
0) of simplexes a
Ea.
That ~ is a mock bundle follows
from [3]. Notice that ~
I
(K x 0)' is a subdivision of ~, and similarly
for ~
I
(K x
1)'.
Now subdivide ~ so that cells
a
x I are subcomplexes
of its base, for
a
E
K. Finally, amalgamate over K x I to realise the
required cobordism.
Remark. The proof of duality shows that there is really no dis-
tinction between bordism and cobordism classes when the base is a mani-
fold; they are represented by precisely the same class of map! The
duality between cap and cup products is also clear from the definitions
using the duality map; i. e.,
If.'W
n
1)
=
If.'(~ U 1)),
etc. See the next
section for connections with transversatility.
General duality theorem
Tq(X)
®
T (X x y)
-+
T + (y)
s
q
s
[~] iZl
[f]
1-+
Tly
o([t]
n
r~
x ly)).
2.
defined by
If.'
{p : E - M} = {p : E ....•M }
!
Theorem 3.2.
If.'
is an isomorphism.
c
Nowlet Y
c
X
C
M be compact subpolyhedra, and denote X ,
yC
for their complements. We can regard Xc, y
C
also as compact
polyhedra by removing the interior of derived neighbourhoods of X and
Y.
Proof.
If.'
is onto: Let f : W - M be a bordism class. We have
to find a mock bundle which 'amalgamates' to W. We can suppose f is
simplicial, and consider the dual cell complex M* to M. Then
f : W - M* is the projection of a mock bundle. This is a consequence
of Cohen's [3; 5.6]. Notice that for a*
E
M*, we have the block over
a* equal to D(a,
0,
which is a manifold with boundary corresponding to
the boundary of a*.
If.'
is
1 : 1.
Suppose ~q,
1]
q/X,
I
K
I
= M are mock bundles, and
Duality Theorem 3.3. There is a natural isomorphism
q _
c c
~ : T (X, Y) T
+
(Y , X ).
n
q
Remarks.
1.
l/>
can be regarded as an extension of the cap
product with [M].
2. (3.3) generalises both Lefshetz duality and Spanier-Whitehead
duality; e. g. , for the latter, take X = M = Sn
(d.
Whitehead
[13]).
30
31

Fig. 3
where a is amalgamation; see Fig. 3.
submanifold. Then we extend to give relative theorems, theorems for
embeddings and for general subpolyhedra. We also give the connection
with block transversality [12].
Let f : W ..• M be a map between compact
pi
manifolds with W
closed, and suppose that N
C
M is a submanifold. Then we can regard
f : W ..• M* as the projection of a mock bundle ~, as in §3. Now ~ can
be subdivided so that N is a subcomplex of the base (this involves a
homotopy of f); then
C
1
(N)
is the restriction of ~ to N, and hence a
manifold by 1. 2, and f is now transverse (in some sense) to N!!
Notice that the proof is easily adapted to give an E-version by
making the diameters of cells of M*
<
E. Further, N can be replaced
by a whole family of manifolds. In fact, the natural setting is where N
is a general subpolyhedron. We now show how to treat relative trans-
verslity in this setting. Let X
C
M be a subpolyhedron. We say that
f : W ..• M is mock transverse to X if f is the projection of a mock
bundle in which X underlies some subcomplex of the base. We write
W
In
X,
or
f
In
X.
For technical reasons, we need a condition on X
=
X n
aM
to
a
get a relative theorem (there are counterexamples otherwise; see 4.2
and 4. 3 below). We say that X is locally collared in (M, X) provided
a -------
that at each point x
E
X there is a neighbourhood in (M, X) which is
a
the product of a neighbourhood in
(aM,
X ) with the unit interval. Local
a
collaring is equivalent to collaring [8; p. 321].
Tq(X, Y) ;:. Tq(N(X), N(Y)) ~ T +q(N(X)- N(Y), N(X) - N(Y))
i*
n
I.
1*
Proof of 3.3. Let N(X), N(Y) be derived neighbourhoods of
X and Y, and define
Ifi
to be the composition
3. By Spanier-Whitehead duality, Tq( ,) is indeed the dual
theory to
pi
bordism.
'To see that
Ifi
is surjective, regard f : M ..•
y
C
as the projection
c
of a mock bundle (in which the blocks might have extra boundary over X)
by Cohen's theorem. Then restrict to X to get a genuine mock bundle.
To see
Ifi
is injective, combine this proof with the second half of the
proof of 3. 2.
4. APPLICATION TO TRANSVERSALITY
We observe that the mock bundle subdivision theorem (together
with Cohen [3; 5. 6]) implies various transversality theorems. We deal
with the simplest case first, the case of making a map transverse to a
Relative transversality theorem 4. 1. Let M be a compact mani-
fold with boundary and X
C
M
a polyhedron with X
=
X n
aM
locally
---------- a
-1
collared in
(M,
X). Let f: W -
M
be a map such that f
aM
=
oW,
and suppose
flow
In
Xo; then there is an E-homotopy of f reI
oW
making f mock transverse to X.
Proof. Suppose f
low
is the projection of the mock bundle UK,
and choose a ball complex L with
I
L
I
=
M extending K, and so that X
is a subcomplex of L. This is done by first extending to a collar via the
product ball complex K x I, and then choosing any suitable ball structure
on M -
I
K
I
x [0, 1) and adjoining the two. Following the proof of 3. 2,
we can suppose that f is the projection of a mock bundle ~ such that
32
33

~ 10M
is a subdivision of ~. Choose a further subdivision
~';1.'
of ~
so that L'
<J
L. Then amalgamating over L, we have ~ , say, over L
1
with
~1
I
K
3;~.
It only remains to observe that the homotopy of p~
takes place within cells of K, and hence can be shrunk to the identity,
and this extends to give a modified homotopy of f by the HEP.
Remark 4.2. In fact, the proof of 4. 1 used only that K extends
to L with X a subcomplex of L. This needs a much weaker condition
on Xa than local collaring. A necessary and sufficient condition is that
the ambient intrinsic dimension [1] of X at X is constant on the in-
a
teriors of balls of K. This is always true if the ambient intrinsic
dimension of X at x
E
Xa equals that of Xa at x.
Example 4. 3. Let X
=
'the letter T' with the top in aM so that
K nX is a I-cell. Then, if f:
oW ....•
aM is not transverse to the mid-
point of K n X, there is no homotopy of f reI aw making f mock trans-
verse to X.
Transversality for embeddings
Now suppose that f: W ....•M is a locally flat embedding (the
condition on local-flatness will be removed later). We say that
f : W ...•.M is an embedded mock bundle if f is the projection of a mock
bundle ~/K, and for each ball
aE
K, we have f
I :
~(a) ...•.a
is a proper
locally-flat embedding (1.e.,
f-1(a)
=
aHa),
and f looks locally like the
inclusion R~_
C
R: for some k, n). We then observe that the sub-
division theorem for mock bundles applies to embedded mock bundles to
yield an embedded mock bundle, and that the homotopy which takes place
in the proof can be replaced by an ambient isotopy (by uniqueness of
collars [5]). Thus Theorem 4. 1 applies to give a relative transverality
theorem for embeddings via an E-ambient isotopy (fixed on aM).
Connection with block transversality
Suppose X
C
M is a compact polyhedron and W is a locally flat
submanifold of M. Recall [12] that X is block transverse to W in M
if there is a normal block bundle
v
/W in M such that X n
E{V)
=
E(vIX
n W). We write X
-.h
W or
xlv.
34
Theorem 4. 4. X
1
W if and only if W
In
x.
Thus Theorem 4. 1 (the version for embeddings) recovers a
strengthened form of [12; 1. 2].
Proof. Suppose X
1-
W; then there is a normal block bundle
II/W
so that X n
E(v)
is a union of blocks. Choose a ball complex K
with
I
K
I
=
M so that the blocks of
v
are balls of K, and so that X
underlies a subcomplex (1.e., simply triangulate the complement of
E(v)
and throw in the blocks of
v!).
Then the inclusion
we
IKI
is
the projection of an (embedded) mock bundle with X a subcomplex of
I
the base. Notice that the restriction of this mock bundle to
E(v)
gives
the Thorn class of
v.
Conversely, suppose W
jp
X by ~/K; then we construct a normal
block bundle
v
on W in M by induction over the skeleta of K so that
it restricts to a normal bundle for
~(a)
in
(J
for each
(J
E
K. This
is an easy consequence of the relative existence theorem for block
bundles [11; 4. 3]. Then X
-.h
v.
Extension to polyhedra
This subsection anticipates §3of Chapter III (manifolds with sin-
gularities). Observe that if f: Y ....•M is a map where Y is a polyhedron,
andwe apply the process of Cohen's theorem [3; 5.6] and regard f as
a 'bundle' over M*, then Cohen's proof shows that the blocks, although
notmanifolds, are polyhedra with collarable 'boundaries'; i. e., if
-1 • -1
(J
E
M*, then f
(a)
is collarable in f
(a).
So we define f :
y ...•.
K to
-1 •
be a polyhedral mock bundle if for each
(J
E
K we have f
((J)
collarable
in
C1{(J).
Then the subdivision theorem works and we thus get a trans-
versality theorem for two polyhedra in a manifold and similar relative
versions and versions for embeddings. In the case of embeddings, mock
lransversality implies transversality in the sense of Armstrong [2].
This is proved by using the collars to construct neighbourhoods of the
form cone x transverse star; compare with the proof of 1. 2. McCrory
[15]has shown that mock transversality for polyhedra is symmetric and
equivalent to both block transversality in the sense of Stone [16] and to
35

transimpliciality in the sense of Armstrong [2].
i
I
inn {} n+l.
r:..
x
II
A x 0 ...•A gIven by
The transversality definition of the cup product
Suppose given ~q, r{;K then for some large m we may assume
EW, E(1])elK
I
x
1
mso that
PeP
are restrictions of the projection
IKI x
1
m...• IKI. Now consider E(~)1]X
1
mE(1]) x
1
m
c
/KI x
1
mx
1
m
.. m
0' 1 0
l'
By mductIvely making ~(a) x I transverse to 17(a) x
1
min ax
1
mx
lID
o
1 0 1
we get a mock bundle E(~) x I~ n E(1]) x I~ ..• K It can easily be seen
that this gives the cup product, using the alternative definition. However
we sketch a proof below connecting the transversality definition with the
restriction to the diagonal definition. This proof has the virtue of
generalising to the more complex situation considered in Chapter V.
Let s denote q-fold suspension. From definitions
q
m m
s2m(~ x 1])
=
sm
Wx t(K x I ) u t(K x I ) x sm(1]) and the total spaces
on the right are transverse without being moved. Let i: IK
I
x
1
2m...•
IKI x IKI x
1
2m be given by i(x, y)
=
(x, x, y). Then applying i* we
get
(t , .•. , t , s) ~ «1 - s)t , ... , (1 - s)t , s).
o non
This determines a natural isomorphism
e(Y) :
nsy ...•sny. There are
also based homotopy equivalences
I/J(Y): ISY/ ...•Y and 1/I(X):x ...•slxl
(see [10; p. 334]). Consider the composition
l/J(nlxl)
0
lelxll °In1/l(x)
I :
Inxl ..• lnslxll ...•lsnlxll ...•nlxl.
Wehave now proved:
Lemma 5.
1.
Suppose X is a Kan based A-set. There is a based
(weak) homotoEY eq~ival~nce
Inxl-nlxl.
Desuspending both sides reveals the coincidence of definitions.
5. THE CLASSIFYING SPECTRUM
Now define an n-spectrum as follows. For each m
€
Z is given
a Kan A-set S and a homotopy equivalence
m
e :
S
-ns
l'
m m m-
n-spectra and A-sets
It
follows from 5.
1
and
[13]
that a cohomology theory h* is defined on
the category of pairs of CW complexes and homotopy classes of maps by
Let ny denote the space of loops on the based CW complex Y,
and let SY denote the singular complex of Y. There is an identification
We give a simple- minded definition of spectra in the category of
A-sets. Basic facts about A-sets contained in
[10]
will be assumed.
Given a based Kan A-set X, we define a based Kan A-set nx as follows, The n-spectra for
pl
cobordism
Define S (PL) as follows. Let R
co
=
URn. Then a k-simplex
m
kook
of
S (PL) is a compact polyhedron X
c
A x R such that
71
I : X - A
m
is the projection of an m-mock bundle over Ak, where
71
is the natural
projection.
Base simplexes *k
€
S(PL)m are defined by taking X
=
¢,
and
face operators are defined by restriction. It follows from the proof of
1.
1
and general position that S (PL) is a Kan complex (compare
[11;
m
o~+
1
a
=
*
o'
The operators
1, ... ,
n, are just restrictions of the
if and only if 0n+l a
=
*nand
O. :
(nX)(n) ..• (nX)(n-l) i
=
0
I "
0.
in
X,
I
36
37

o
k k k+1 k()
1 1
2.3]).
Dehne e : 6.
-+
6. byes
=
2S
+ 2vk+l'
ek: !j(PL)(k)
-+
(Ug(PL) )(k) defined by
m m-1
00
k
x
oct
00
R
eI
A
XR
X
c
L\.
x •••..•
k+ 1 .
Then we have gives a mock bundle
11
/CK such that s ~/K x I is isomorphic to the
#
amalgamation of the pull- back
1f
11
by the pinching map
1f:
(Kx I)' - K,
where (K x I)' <l K x I and
1f
is simplicial.
We have proved:
e : g(PL) - U!j(PL) 1 is then a based 6.-map.
m m m-
Proposition 5.
2.
equivalence.
e : g (PL)
-+
ug (PL) 1 is a homotopy
m m m- ------
Theorem 5.3. The cohomology theory {Tq,
0 }
coincides for
-----~~-~ q -----
polyhedral pairs with the cohomology theory defined by {g(PL) ,e
1.
"--"-----=-
--"c=..-_~~ ~
m m
REFERENCESFOR CHAPTER II
where []6. denote Jl-homotopy classes. Then by general position (com-
pare
[11;
§
2]), <I>
is a bijection, and it follows from
[10]
that
<I>
induces
[11]
a natural bijection
Proof. e is injective so we have to describe a deformation
[1]
m
retraction of ug (PL) on e (g (PL) ). This can be thought of as
m-l m m
'sliding to the half-way level'. More precisely, suppose given a map
[2]
A
0
-+
ug (PL) whose boundary goes into e (g (PL) ). Then glue
n
1
m-l m m
the'simplexes together to form a mock bundle UCA
0
embedded in
[3]
n,l
CA
0
x Roo and empty over cone-point and base, where CA
0
denotes
n 1 n,l
a co~e on A
0
with cone-point last. Then E(~) lies over the half-way [4]
n,l
level in C
oA
0
and we homotope E(~) reI boundary into the pre- image
n,
l'
of the half-way level over CA
o'
Then, using an identification of
[5]
n,l
C6.nwith CA
0
x I and general position, we get an n-simplex of
n,
1
th
Ug(PL) whose restriction to the i face lies in em(g(PL)m)' The
[6]
m
deformation then follows from [10;
6.
3]. [7]
Nowlet K be an ordered simplicial complex, and f : K-g (PL)m
a 6.-map. Then we can form an m-mock bundle UK by gluing together [8]
the images of simplexes of K, and since the base complex gives the
empty bundle, we get a function [9]
m
<I>:
[K, L; g (PL)m'
*]Jl -
T (K, L),
for polyhedral pairs.
Finally, we notice that the suspension isomorphism essentially
coincides with the function em' More precisely, em
0
f: K-U(g(PL)m_ll
38
[10]
[12]
[13]
E. Akin. Manifold phenomena in the theory of polyhedra. Trans.
A. M. S., 143 (1969), 413-73.
M. A. Armstrong. Transversality for polyhedra. Ann. of Math.
86 (1967), 172-91.
M. M. Cohen. Simplicial structures and transverse cellularity.
Ann. of Math.
85 (1967), 218-45.
M. M. Cohen and D. P. Sullivan. On the regular neighbourhood
of a two-sided submanifold. Topology,
9 (1970), 141-7.
J. F. P. Hudson and E. C. Zeeman. On combinatorial isotopy.
Publ. I. H. E. S.,
19 (1964), 69-94.
H. Morton. Joins of polyhedra. Topology,
9 (1970), 243-9.
F. S. Quinn. A geometric formulation of surgery. Ph. D. thesis,
Princeton University,
1969.
C. P. Rourke. Covering the track of an isotopy. Proc. Amer.
Math. Soc.,
18 (1967), 320-4.
C. P. Rourke. Block structures in geometric and algebraic
topology. Report to I. C. M., Nice,
1970.
C. P. Rourke and B. J. Sanderson. 6.-sets I. Quart. J. Math.
Oxford,
2, 22 (1971), 321-8.
C. P. Rourke and B. J. Sanderson. Block bundles I. Ann. of
Math.,
87 (1968), 1-2B.
C. P. Rourke and B.
J.
Sanderson. Block bundles
n.
Ann. of
Math.,
87 (1968), 256- 78.
G. W. Whitehead Generalised homology theories. Trans. A. M. S.
102 (1962), 227-83.
39

[14]
[15]
[16]
40
E. C. Zeeman. Seminar on combinatorial topology. 1.H.E. S.
and Warwick, 1963-66.
C. McCrory. Cone complexes and
pl
transversality. Trans.
A. M. S. 207 (1975), 1-23.
D. Stone. Stratified polyhedra. Springer-Verlag lecture notes,
no. 252.
III· Coefficients
In this chapter we give a geometric treatment of coefficients in
oriented pl bordism theory. The definition (although not all the
theorems) extend to other geometric homology theories and this extension
will be covered in Chapter IV, as will the extension to cobordism (mock
bundle) theories. Application to general homology theories and con-
nection with other definitions of coefficients will be covered in Chapter
VII.
There are two good definitions of coefficients:
1. For a short resolution
p
of an abelian group G we define
coefficients in
p
by labelling with generators and introducing one stratum
of singularities of codimension 1 corresponding to the relations (see §1).
2. We allow labelling by any group elements, and singularities
corresponding to any relation and then, in the bordisms, allow singulari-
ties of codimension 2 corresponding to 'relations between relations
I
(see §3).
Definition 1 is very simple geometrically while definition 2 is
functorial in G. To prove equivalence of the two definitions involves a
further definition, for longer resolutions (in §2). The basic geometrical
trick is resolution of singularities and appears in the proof of the universal
coefficient sequence in §2. The universal coefficient sequence itself can
be seen as the measure of the obstruction to resolution of the final singu-
larity. In §3 it is seen that the universal coefficient sequence is natural
for G; consequently by [3] it splits for a large class of abelian groups,
including all groups of finite type.
In §4we show how to regard the product of a (G, i)-manifold with
a (G', i')-manifold as a (G®G', i+i')-manifold and thus define a cross
product for bordism with coefficients.
41

In §Sis the Bockstein sequence and in §6we observe that, if
G is an R-module, then n*(-, -; G) inherits an U*(Pt.; R)-module
structure in a natural way. Using Dold [1] we then have a spectral
sequence Tor (U (-, -; R), G)
o-?
U*(-, -; G).
p q
We use the convention - ~
+
for inducing orientations on the
boundary of I-manifolds, and in general use the 'inward normal last'
convention.
1. COEFFICIENTS IN A SHORTRESOLUTION
Let F be a free abelian group with basis B. We define co-
efficients in F by labelling with elements of B, i. e. define an F-
manifold to be an (oriented
pi)
manifold each component of which is
labelled by an element of B. We write (M, b) or Mi&>b for M labelled
by b. It is easy to see that bordism of F-manifolds defines a bordism
theory n*(,; F) and that U*(X, A; F) ~ U*(X, A) i&>Ffor the pair (X, A).
Nowlet G be a general abelian group and
p
a short resolution
of G.
p
comprises a short exact sequence
1/>1 E
O •..• F •..•
F ...•
G •..• O
1 0
where F and F are free abelian groups with bases Band B .
o
1 0 1
The elements of
Bo
are the generators of G and those of
B
1
are the
relations for G with respect to the resolution
p.
We will define co-
efficients in
p
by starting with U*(,; F ) and 'killing' the elements of
o
U (pt.; F ) which correspond to the relations:-
o
0
Let r
E
B then
I/>
(r)
E
F and can be written uniquely in the
1 1 0
form
L
O'.b. where
0'.
is an integer and the sum is taken over elements
111
b.
E
B. The element L(r,
p) E
U (pt. ; F ) is the union of
10' I
points
1 0 0 0
i
labelled by b. and oriented
+
if
0'.
>
0 and - if
0'.
<
0, where the
1 1 1
union is taken over all elements b.
E
B .
1 0
A p-manifold of dimension n is a polyhedron P with two strata
P
=:J
S(P), labellings and extra structure such that
1. P - S(P) is an F -manifold of dimension n.
o
2. S(P) is an F -manifold of dimension (n - 1).
1
42
3. For each component
(Q,
r) of S(P) there is given a regular
neighbourhood N of
Q
in P and an isomorphism
h: N ...•Q
x
C(L(r,
p))
where C(X) denotes the cone on X and h carries
Q
by the identity
to
Q
x (cone point).
4. h is an isomorphism of F
0
-manifolds off
Q.
Intuitively, a p-manifold is a manifold labelled by generators for
G with codimension 1 singularity labelled by relations. Moreover the
sheets and orientations of P near S(P) give the relation labelling the
singularity.
Examples 1. 1. 1. G
=
F a free abelian group and F
=
0
1 '
F
0
=
F,
Bo
=
B.
Then a p-manifold is precisely an F-manifold.
2. G
=
Z and we use the usual presentation
n
Xn
O-Z ...•
Z ...• Z ...•
0
n •
Then B
=
{Il, B
=
{Il and so the labels give no information
o
1
and can be suppressed. L(I,
p)
is the union of n points (all oriented
+).
Thus a p-manifold is a manifold with a codimension 1 singularity, which
has a trivial neighbourhood of the form C(n points). Moreover the
orientations of the n sheets all induce the same orientation on the singu-
latiry. This is precisely Sullivan's description of 'Zn-manifold' [4].
See Fig. 4.
Fig. 4. Part of the neighbourhood of the singularity in a 'Z -
manifold'
3
43

3. G
=
Z
00'
We describe the resolution by specifying the
p
generators and relations. The generators are 'liP',
'liP
2" •••
and
the relations, as elements of F , are (P('l/pi,) - 'liPi-l,), i= 1, 2 ...
o .
and p('liP'). Thus the sheets are labelled 'l/pl, and the singularities
occur where p sheets labelled 'l/pi, merge into one sheet labelled
'l/pi-l, or where p sheets labelled 'l/p' merge together. Orienta-
tion has the obvious compatibility. See Fig. 5, in which p = 5:
Fig. 5
4. G
=
Q,
the rationals. Generators 'l/n', n
=
1, 2, .,. and
relations p('l/n') - 'l/q', where n = pq and p is the smallest prime
occurring in n. The picture is similar to the last one.
5. In all the cases above we have chosen the most natural resolu-
tion for G. Other resolutions also give rise to a notion of 'G-manifold'.
In the example below, a, b, c, d are the generators of the copies of Z
indicated:
ZGlZ-ZGlZ--+-Z
3
b-l
c~a-2b
d
l ••
a+b
Here a p-manifold has two sorts of sheet: a-sheet and b-sheet.
Two b-sheets can merge into an a-sheet and an a and b sheet can
44
merge together. This notion gives the same bordism theory as the
notion of
'Z3
-manifold' in Example 2, as we will show in §3.
There is a natural notion of p-manifold with boundary and we thus
have a bordism theory n*(,;
p).
That this theory is a generalised
homology theory follows from the proof given in II 3.1 for T*( ,).
(The crucial fact is that the regular neighbourhood of a polyhedron in a
p-manifold can be given an essentially unique structure as a p-manifold )
We now turn to the universal coefficient theorem for p-bordism. In §2
we will prove the theorem in the general case (for longer resolutions)
and here we will content ourselves with the statement and a sketch
proof of this (simpler) case, with stress on the geometry of the situation.
Theorem 1.2. Let
p
be a short resolution. There is a short
exact sequence, natural in (X,
A):
l
s
0'" n (X, A)
®
G'" n (X, A;
p) -
Tor(n 1(X,
A),
G)'"
O.
n n n-
Sketch of proof. (For full proof see 2. 5.) The spaces (X, A)
play no role in the proof of the theorem and we will ignore them.
The notation is intended to suggest that
l
is the 'labelling' map
and s is 'restriction to the singularity'.
Description of
l
Using the description of n
®
F as manifolds labelled by ele-
n
0
ments of B
0
(see start of this section) we have a 'labelling' map
II :
nn
®
F
0 •••
nn(p)· Now
II
is zero on relations. This is seen as
follows. Let r be a relation and [M] En. Consider the labelled
n
manifold M x L(r,
p)
x I with each copy of M x
{O}
identified to-
gether and this end labelled by r. This constructs a p-bordism of
II
([M], r) to zero. See Fig. 6. It follows that
II
defines a mono-
morphism
l :
n
®
G'" n
(p)
n n'
45

W
=
M
x
I
U
W
x
C(L(r,
p»
1
r
Exactness at 0n(P)
Order 2 is trivial (an F -manifold has no singularity). To see
a
exactness, suppose M is a p-manifold with S(M) bordant to zero as an
F -manifold by a bordism W. Construct the bordism
1
Description of s
Let P be a p-manifold with singularity S(P). S(P) is an F
1-
manifold, moreover the map
®lJll
n ®F -
n
l®F,
n-l
1
n- a
can be described on generators as product with L(r,
p).
Hence the image
of S(P) in n ® F is represented by
au,
where U is a regular
n-l a
neighbourhood of S(P) in P. This is bordant to zero in nn_l ® Fa
since it bounds
cl
(P - U). Thus the 'singularity' defines a map
.n
(P)- Tor(n G) and it is surjective by reversing the above
s. n n-l'
argument.
M
x
I
W
W x C(L(r,
p»
Fig. 7
2. COEFFICIENTS IN A LONGER RESOLUTION
where W is a typical component of W (labelled by r) and the union
r
identifies the obvious subset of M x {l ) with
aW
rx C(L(r,
p)). WI
is a p-bordism of M to an F -manifold. See Fig. 7.
a
In this section we will generalise the construction of §l to give
coefficients in resolutions of length
:5
4.
Let G be an abelian group. A structured resolution
p
of G
consists of
(a) a free resolution of G of length
:5
4
1Jl
3
1Jl
2
1Jl
1E
O-F
3
-F
2
-F
1
-F
o
-G-0
(b) a basis, BP, for each F (p
=
0, 1,
2, 3)
P
(c) for each bP
E
BP(p
=
1,
2, 3) we are given an unordered word,
w(ll), representing the element IJlp(bP). Precisely w(ll) is a finite set
(ofpairs)
M
Fig. 6
M
x
L(r,
p)
46
47

where the structure is that induced from
p.
Clearly
Po
=
p.
For each quadruple (G, p, p, n), where G is an abelian group,
p
is a structured resolution of G; p, n are integers such that
-1 ~ P ~ 2; n
2:
0, we shall construct
(a) a class of pl isomorphism types of polyhedra with extra
structure, called the class of
(p,
p, n)-manifolds.
(b) a class .cPof
(p,
p-I, p)-manifolds for p
2:
0, called the
class of (p, p)-links.
We start by defining a
(p,
-1, n)-manifold to be an Fo-manifold
and in general
(p,
p, n)-manifolds will be defined from
(p,
p-I, n)-
manifolds by 'killing' the class .cp. The precise definition (an extension
of that in §I) is given at the end of the construction of the classes .cp.
The
(p,
2, n)-manifolds will be called simply
(p,
n)-manifolds.
Construction of .cP
For diagrams we refer to the examples given later. To construct
l
.cO let b
1E
B\
w(b
1)
=
Z; o.b? Give the set {b~, ... , b~} the dis-
i=I
1 1
crete topology and each point b. (i
=
1, ... , l) the orientation 0.. The
1 1
resulting polyhedron will be called the a-link associated to b
1
in
p
(or generated by b
1
in
p)
and will be written L(b
1,
pl.
We define
the class .cO to consist of all polyhedra L(b
1,
p)
with b
1E
B
1,
i. e.
.cO
='
{L(b
1,
p) :
b
1E
B
1
J.
Nowconsider the join of L(b
1,
p)
and the
point b1, written b
1
L(b1,
p),
and give b
1
L(b
1,
p) -
b
1
(=
the open
cylinder over L(b
1,
p»
the orientation -
-+-
+
(arrow departing from
'-' sign). The join b
1
L(b
1,
p),
with the above orientation off b
1
and
with the orientation
'+,
on b
1,
will be referred to as the (oriented) cone
over L(b
1,
p)
with vertex b
1•
In general it happens that different
elements of B
1
may give rise to the same a-link. Therefore, over the
same link, there may be different cones, corresponding to different
vertices generating the link.
1 2 2 2
l
(b
2) 1
To construct .c , let b EB ; w(b )
=
2: o.b.. Construct
i=I 1 1
the a-link L(b
2:
p )
as in the previous case. Each b~ generates a
1 1
Q-link L(b~,
p)
(i
=
1, ... ,
l);
consider the space ~ o.[b~L(b~,
p)]
=
L
1 i=11 1 1
where b~L(b~,
p)
is the cone as defined above and O. changes all the
1 1 1
orientations present in this cone iff 6.
= '-'.
Let
Vi
= ~0.w(b~)
=
1 1 1
lbO, ... , b~ with the appropriate signs
L
Then
L
= ~
o.L(b~,
p)
is
1
i=I1 1
obtained by giving
Vi
the discrete topology and orientations according
to the signs. Therefore the cancellation rule c(b
2)
gives a canonical
way of joining the points of
L
in pairs by plugging in oriented I-disks.
Precisely suppose 0jb~ is paired with Ok1\.. Then OJ'" Ok and we
insert a I-disk [b;, ~] with orientation given according to the rule
'arrow departing from
,+,
sign'. Moreover we label the I-disk by the
unique element b;
=
b~
E
BO• The object which is obtained from L
through the above identifications in
L
is called the I-link associated to
b
2
in
p
(or generated by b
2
in
p)
and it is written L(b
2,
pl.
The
class of I-links, .c
1,
is defined by
£1
='
{L(b
2,
p) :
b
2E
B
2}.
1m ¢k
p
is the structured resolution of
k
t/>k
•.•. Fk •.•.1m ¢k .•.•
a
such that ~ .0.b~-1
= "
(bP) where the sum denotes the element of Fp 1
1 1 1 "1'p , -
obtained from w(bP) by adding up all the coefficients belonging to the
same element in BP-l.
(d) For each bP
E
BP(p
=
2, 3) we are given a 'cancellation rule'
defined as follows. Let w(bP)
=
{Oibr 1 : i
E
I(bP)). Then order two
of
p
implies that the formal sum Z;.0.w(b-?-1) is an unordered word
1 1 1
representing the zero element of Fp-2 in terms of the elements of
BP-2The effect of O. is an inversion of sign iff 01,
= '-'.
• 1
The given cancellation rule consists of a procedure for pairing off
the letters of 2;.o.w(b-?t-1) in F 2' Precisely c(bP) is a partition of
1 1
p- .
0_
2 •
0-
2 .
the letters of LiOiw(bf-1) into pairs of the form (Ojlr , 0klr ) w1th
O. '" Ok'
J
For the sake of simplicity we may write the set I(bP) in the form
{I, ... ,
l : l
=
l
(bP)
J.
If a ~ k ~ 3 then
48
49

(d) h is an isomorphism of
(p,
p-l, n)-manifolds off V.
Examples 2.2. 1. A short resolution gives rise to an obvious
structured resolution (the cancellation rule gives no information in this
case). Then the notions of p-manifold defined in §l and §2 coincide.
Remarks 2.1. 1. It is obvious how to give a p-link the required
extra structure to make it into a
(p,
p-l, p)-manifold (and this completes
.all the definitions).
2. In all the above definitions we have only used the fact that
the resolution
p
has order two.
3. The definition extends to yield
(p,
n)-manifolds with boun-
dary in the obvious way.
¢2 ¢1
- F Ker e: - FZ - Z - 0
3 3
1/>3
p :
0 - Ker
I/> -
F Ker ¢
3 2 1
b EKer
¢
be such that
2
2.
Let
W is an oriented manifold,· are defined as in the previous cases.
We now define
(p,
p, n)-manifolds (without boundary) inductively
on p as follows. A
(p,
p, n)-manifold is a polyhedral pair
M:) S(M)
with labellings and extra structure such that
(a)
M - S(M)
is a
(p,
p-l, n)-manifold
(b)
S(M)
is an FpH-manifold
(c) (Trivialised stratification condition) For each component V
of
S(M)
labelled by bP+lE BP+\ there is given a regular neighbourhood
N of V in M and an isomorphism
w(b
3)
=
_b
2
+ b
2
+ b
2
123
w(b
2)
=
_b
1
+ b
1
EF Ker e:
1 11 12
w(b
2)
=
b
1_
b
i
+ b
1
EF Ker e:
2 21 22 23
w(b
2)
=_b
1
+b
1
+b
1
EFKere:
3 31 32 33
w(b
1)
=
bO+ bO
11 1 2
1
° °
w(b )
=
-b - b
12 1 2
It is clear that it is the given cancellation rule in
p
that makes
the construction well defined. Different rules may give rise to com-
pletely different links.
Wethink of L(b
2,
p)
as a one-dimensional stratified set in which
the intrinsic j-stratum
(j
=
0, 1) consists of a disjoint union of j-disks,
each disk is oriented and labelled by one element of Bl-j; the O-stratum
is the O-link generated by b
2
in
p .
I
It is clear how to define the cone b
2
L(b
2,
p):
topologically
2 2 . 2. 2
b L(b ,
p)
IS the usual cone over L(b ,
p)
wIth vertex b , the subcone
over the O-stratum of L(b
2,
p)
is given an orientation outside b
2
as
in the previous step; the subcone over the I-stratum has the orientation
given by the cartesian product:
(I-stratum) x [-; +) where [-; +) is the half open I-disk oriented
'from - to +'. Finally the vertex b
2
has the orientation +. Each
stratum of b
2
L(b
2,
p)
is labelled in the obvious way. Now, as before,
it may wellhappen that different elements in B
2
generate the same
I-link and therefore over the same link there may be different cones.
Nowsuppose W is an oriented manifold, then we can form the
topological product W x b
2
L(b
2,
p).
From now on we think of the above
product as having the following additional structure: three intrinsic strata,
namely W x b
2
W xL, W xL, L. being the intrinsic j-dimensional
, °
I
J
stratum of L(b
2,
p)
(j
=
0, 1); a labelling on each stratum obtained from
the labelling of the second factor; the product orientation on each stratum.
We are now left with the case p
=
2. Let b
3
EB
3•
Consider
the I-link associated to b
3
in
p
and construct a trivial normal bundle
I
system with base L(b
3,
p )
as follows. If 6.b~ is a vertex of
III
L(b
3,
P )
then put 6.L(b~,
p)
as the fibre at that vertex. The part of
III
L(b
3,
P )
which remains unclothed consists of a disjoint union of closed
I
I-disks. Let D, labelled by b
i
EBI, be one of such disks. The re-
striction of the normal bundle to aD is aD x L(b\
p);
therefore we can
extend the bundle by plugging in D x L(b
1,
pl.
As a result of clothing
the I-stratum of L(b
3,
P )
we are left with a polyhedron, whose boun-
1
dary consists of I-spheres, labelled by elem.ents of BO. Then plug in an
oriented labelled 2-disk for each sphere and get the required link L(b
3,
pl.
The cone b
3
L(b
3,
p)
and the product W x b
3
L(b
3,
p),
where
50
51

52
w(b
1)
= bO + bO
21 1 2
'(b
1)
= bO+ bO + bO + bO + bO
\\ 22
°
11 12 21 22
w(b
1)
=bO +bO +bO
23
°
1 2
w(b
1)
= bO + bO+ bO
31
°
1 2
w(b
1)
= bO + bO + bO + bO + bO
32
°
11 12 21 22
w(b
1)
= -b
° -
b
°
33 1 2
bO =bO =bO EFZ . bO =bO =bO EFZ . E(bO)= 0 EZ .
11 12 1 3' 21 22 2 3'
°
3'
E(bO)=
1
EZ ; E(bo) = 2 EZ
1 3 2 3
Fig. 8
Figure 8 shows the construction of L(b
3,
p),
the cancellation rules
being suggested by the diagram itself.
CP2 CPl
3. P :
0 - Ker
cp -
F Ker
cP -
F Ker e - FZ - Z -
O.
2 2 1 5 5
Let b be a basis element of F Ker
CPl'
Suppose
w(b
2)
= b
1
+ b
1
+ b
1
+ b
1
1 2 3 4
W(b
1
1)
= bO + bO + bO
11 12 13
w(b1) = _be + bO + bO
2 21 22 23
w(b1) = _be _ bO + bO
3 31 32 33
w(b
1)
= _be _ bO _ bO
4 41 42 43
° ° ° °
E(b
11)
=
E(b
43)
=
2; e:(b
1)
=
e:(b
4)
=
3;
e(bo )
=
e:(bo )
=
e(bo )
=
E(bo ) =
O·
13 21 33 41 '
0) 0)
°
0)
e(b
22
=
E(b
32
=
1;
e:(b
23)
=
E(b
31
=
4.
Fig. 9
Figure 9 shows two possible links associated to b
2
in
p
corres-
ponding to different cancellation rules. This completes the examples.
Now let M be a
(p,
n)-manifold and M' a
(p,
n')-manifold. An
embedding f : M' - M is a locally flat stratified embedding between the
53

Consider the following spaces:
Proof.
M x 1', where l' = [0, -1];
SR = any bordism between -SM and -S:
assume that SR consists of a set of equally labelled
components, with label, say, bPEBP;
v(-SM)
=
normal bundle of -8M in -M=MX {-I};
SR
x
L(~,
pl.
R
=
SR x L(bP,
p) ~))
M
X
I'
Lemma 2. 6. Suppose that [SM]
=
[8] as Fp-manifolds. Then
M
is bordant to a
(p,
n)-manifold Q, whose last stratum is S, by a
bordism R whose last stratum is still in codimension p.
Description of the map s
Proof. First of all we anticipate that the proof consists of
geometrical arguments involving only (p, n)-manifolds and their strati-
fications. In the constructions, the maps into
(X,
A) do not play an
essential role and so, for the sake of simplicity, we shall assume
(X,
A)
=
(point, ¢).
2.
We
have:
v(-SM)
C
M x {-I};
v(-SM)
C
SR x L(~,
pl.
So we can form
theidentification space:
Proposition 2.4. {nn ( , ;
p)}
=is=--=a~g",:e=-=n=-=e:..:.r--=a:..:.li=s--=e:..:.d--=h-=-o:..:m=-=o-=-lo::."gy,,-,,-..:.:th-=-e:..:o:..::r.:!-y
on the category of topological spaces.
It involves a resolution of singularities. Let M be a closed
(p, n)-manifold. We are going to show that there exists a
(p,
n)-manifold
M,
bordant to M and having no singularities in codim
>
1. Let SM be
Proposition 2.3. 1.
!!
M is a (p, n)-manifold, M x I has a thelast stratum of M. Then, by definition of
(p,
n)-manifold, SM is
natural structure of
(p,
n+I)-manifold, obtained by crossing the structure .anFp-manifold, where SM has codimension p. We need the following:
of M with I, it is clear that a(M x I) ~ M
u -
M
u
aM x 1.
If M, M' are
(p,
n)-manifolds and M , M' are
(p,
n-l)-
_ -- 0gO--
submanifolds of
OM,
aM' res_.p~e:..:c:..:t-=-iv:..:e:..:l,,-y--=s::..:u::.:c:..:.h=--t.:...h.:...a_tM ~ -M', then
_______ - 0
°
MUM' is a (p, n)-manifold with boundary isomorphic to
g ---"__
--'0--__
Cl(aM\M
U aM'\M').
o
g
0
3. Let M be a
(p,
n)-manifold and
X
C
M. Let N be a
regular neighbourhood of
X
in M. Then N can be given the structure
- --
~~ (p,
n)-manifold in an essentially unique way.
The proof of 2. 3 is left to the reader. There is an obvious notion
of a singular
(p,
n)-manifold in a space and thus we have the bordism
group n
(X,
A;
pl.
The following proposition follows directly from
n
proposition 2. 3, using the proof of II 3. 1.
underlying polyhedra, which is compatible with the labelling and the
trivialisations. If n' = n, then f may be orientation preserving or
orientation-reversing. In the following, unless otherwise stated, a co-
dimension zero embedding will always be assumed to be orientation pre-
serving. A submanifold of a (p, n)-manifold M is a subset M
C
M
----- °
together with an embedding f : M'
'+
M (of p-manifolds) such that
f(M')
=
M. If M is a
(p,
n)-manifold, -M denotes the
(p,
n)-manifold
°
obtained from M by reversing all the orientations; (p, n)-manifolds
have the following properties.
The universal coefficient sequence
Proposition 2. 5. For each integer n
2:
0 and each pair
(X,
A)
there exists a short exact sequence
hich provides the required bordism.
If SR has many labelling elements, we perform the above con-
truction simultaneously on every set of equally labelled components.
rhe last stratum of R is SR and has codimension p. See Fig. 10.
which is natural in
(X,
A).
54 55

M
x
II
The whole manifold
is
R
II'
Fig. 10
Remark 2. 7. If we can choose S =
¢
then the above constructio
gives a resolution of the low dimensional singularities of M. In other
words, when the low dimensional singularities of a
(p,
n)-manifold M
bound (in a labelled sense), they can be resolved by means of a bordism
having the same kind of singularities as M. (Cf. proof of exactness in
1. 2. )
is exact and so there exists an element [SW]
€
n
I8iF such that
n-p p+l
~P+l[SW] = [SM], Suppose first that SW is a set of components all
labelled by bP+1
€
BP+1, We can always reduce to the case
8M = SW I8iw(~+l), because if SM is only bordant to 8W I8iw(~+l):
8M - SW I8iw(~+l), then, by Lemma 2. 6, M can be replaced by another
(p,
n)-manifold M such that:
(a)
M
is bordant to M by means of a bordism with singularities
up to codimension p only
(b) SM= SW I8iw(bP+I),
Therefore assume 8M = SW I8iw(bP+I), If SW-
¢
we are reduced to the
case of Remark 3 and we know how to solve the singularities then, So
assume SW'"
¢
and take the following spaces:
M x I, where 1=[0, -1]
-(SW x bP+lL(bP+l :
p))
v(-SM)
= normal bundle system of -8M in M x
{-I}
= -M.
Then we have
v(-SM)
C
M x I and
v(-SM)
C
-(SW x bP+1L(bP+1,
p).
The identification space
W=_(SWXbP+lL(bP+l,
p) '-------"
MXI
v(-S(M)
So now we have a well defined procedure to solve the singularities
of a
(p,
n)-manifold M, stratum by stratum, starting from the last one
and going up by one dimension each time, until we are left with a
(p,
n)-
manifold,
lVi,
which is bordant to M and has singularities 8M in co-
dimension one at most. In general we cannot solve SM as above, because
Proof of Proposition 2. 5 (continued). Now let us look at the realises a bordism between M = M x
{D}
and a
(p,
n)-manifold M'
image of [SM] through the morphism whose last stratum has dimension n - p + 1. The singularities 8M have
_ id I8iI/>p been resolved up to bardism. In general, if SW is of the form
I/> :
n
I8iF -
n
I8iF . nP+1
P n-p p n-p p-l SW= ~k(SW)k I8i
K '
then one performs the above construction simul-
We have for each component V I8i
If
C
SM ~ ([V.] I8ib.) = [V.] I8iw(llltaneously on all terms (SW)kI8iiJ'+I and gets the desired manifold M'.
, i i
'pI I 11
K
[V.]
&> ~
.6.b~-1 =
L
.[6.V.] I8i})\J-I: this is nothing else than the bordism See Fig, 11.
cl~s of Jt~e~oundar~ ~f ~he cdmPlement of a regular neighbourhood of
V
We remark that in order to get rid of singularities in codimension
in the (n-p+ I)-stratum of M. Therefore the image of [8M] is the bar-
p
we have used bordisms, which have singularities up to codimension
dism class of the boundary of the complement of a regular neighbourhood
P
+ 1.
SM in the (n-p+ I)-stratum of M and, as such, is the zero element of
n
I8iF I' Now, because p> 1, the sequence:
n-p p-
~p+l I/>p
n
I8iF -
n
I8iF -
n
I8iF
n-p p+l n-p p n-p p-l
56
57

Proof. The intrinsic stratum of M in codimension p
+
1 is
a closed polyhedron, because the singularities of
oM
are in codimension
p at most. Therefore the above construction for solving the singularities
can be applied, essentially unaltered, to solve the codimension p
+2
stratum of M,
oM
by a bordism with boundary, W, OW. No new singu-
larities are created along the boundary during this process.
Proof of Proposition
2.5
(continued). Each M
E
[M] deter-
- p
mines an element of Ker
cp ,
namely [8M]. This assignment depends on
I
the representative M; in fact, if M - M, then
s1\lt
may be bordant to
_ I. I
8M by means of a bordism, V, with singularities and therefore in general
[8M
I] '"
[8M]. However, by Lemma
2.
8, the singularities of V can be
assumed to have codimension one at most and we can certainly write:
[8M]
=
[8M
I]
+
[N] where [N]
E
1m
~2'
Thus there is a well defined
map:
I
-----M'
-~
/
The whole manifold is W.
s :
n
(P) -
n
[M] ~
H
(p, n
1)
I
n-
[8M] +
im ~ .
2
is not necessarily exact. However ?>[8M]
=
0 in
n
1
®
F because
I·
n-
0
it is the bordism class of the boundary of the complement of a regular
-
-
neighbourhood of 8M in M.
The next lemma is important in what follows.
Lemma
2.8.
Suppose that M is a
(p,
n+1)-manifold with boun-
dary
oM.
Then, if
oM
has singularities up to codimension p and M
has singularities up to codimension p + h, there is a
(P,
n)-bordism
with boundary, W, OW,between M,
oM
and N,
oN
such that
(a) W has singularities up to codimension p
+
h
+
1 at most,
(b) OW has singularities up to codimension p,
(c) N has singularities up to codimension p
+
1.
Fig. 11
the sequence
It is straightforward to check that s is a morphism of groups.
s is an epimorphism. Take [V]
+
im?>
E
H
(p,
n ).
_ ------ 2 I
n-1 '
~I
[V]
=
O. 8uppose V constantly labelled by b
lE BI;
then
V
®
w(b
l)
bounds in
n
1
®
F , i. e. V
®
w(b )
=
aV.
Take a copy of
n-
0 I
V
and label it by b
l;
aV
consists of a number of copies of V (non
constantly labelled in general); identify each copy with V
®
b
l•
V,
with
the above identification on its boundary, becomes a
(p,
n)-manifold
W
with singularities in codimension one only. Therefore, to each manifold
V,
representing an element in H
(p,
n
1) we are able to associate a
I
n-
(p,
n)-manifold
W,
representing an element in
n
(p),
such that
__ n
s[W]~ [V]
+
im
cp
2'
In fact, if
W'
E
[W]
we can assume, by Lemma
2.
8,
that W' has singularities 8W' in codimension at most one and that there
exists a bordism N:
W' -
VI
with singularities 8N in codimension two
at most. Then
[sW] -
[8W']
=
¢2[8W] and so s is an epimorphism.
~I
n
®F -
n
1®F
n-1
I
n-
0
~2
n
1®F -
n-
2
58 59

Description of the map
l
running:
Define a map
I:
n
®
F ..• n
(p)
by
l[M]
=
[M];
l
is a well
non
defined homomorphism; so we have the sequence
l/JI
l
n
®
F ..• n
®
F ..• n
(p).
n
1
non
H
(p, n
(-))=}
n*(-;p).
p q
This spectral sequence collapses to the universal coefficient for-
mula when
p
is exact.
l
fiJ
=
0 because let [W]
=
lli>
[M] and suppose M constantly labelled
1 1
by bIEBI. Then take M x bIL(bI,
p)
and observe that a(Mx bIL(bJ,P
is bordant to W. So stick the two bordisms together and get a bordism
of W to the empty set by means of a
(p,
n+1)-manifold with codimension
one singularities.
Assume now
l([M])
=
O.
Pick a representative V of
l([M]):
there exists a bordism, N, of V to
l/J
such that N has singularities
SN in codimension one at most. We claim that ~ [SN]
=
[M]. In fact
1
remove from N a regular neighbourhood of SN in N to get the required
bordism between M and
Ii>
(SN). Thus we have proved that the sequence
1
above is exact; which is enough to ensure the existence of a monomor-
phism
l :
H
(p,
n ) ..• n
(P)
induced by
l.
ann
Now it only remains to prove exactness at n
(P).
---- n
sl
=
0 :
sl[M]
=
0, because [M] has no singularities; hence
sl
=
O.
3. FUNCTORIALITY
The classes of links constructed in Section 2 summarize the whole
structure of the resolution
p
geometrically. Therefore, from now on,
we refer to
p
as a linked resolution.
If
P,
p'
are linked resolutions of G, G' respectively, a chain
map f :
p .•• p'
is said to be a map of linked resolutions (or simply
linked map) if f is based and link preserving, i. e. :-
(a) f(bP) EB'P for each bPEBP.
(b) Let bPEBP. If we relabel each stratum of the link
L(tP,
p)
according to f and if f(L(bP,
p))
denotes the resulting object,
then f(L(bP,
p))
=
L(fbP,
p').
So there is a category,
e,
whose objects are linked resolutions
E
P •••
G and whose morphisms are linked maps. If Cib* is the category
of graded abelian groups, we have the following
Ker s
C
im
l:
let [M] E n
(P)
and assume, without loss of
n
generality, that M has codimension one singularities SM; s[M]
=
0
- p
means that [SM]
E
im
l/J.
But then SM can be re-solved up to bor-
2
dism; therefore [M]
=
[M'] where M' is without singularities and hence
determines an element of n
®
F whose image through
l
is [M].
n a
Thus Ker s
C
im
l
=
im
t.
The proof of the proposition is now complete.
Remark 2. 9. We have seen how the exactness of
p
is used in
Proposition
3.1.
n*(X, A;
p)
is a functor
e ..•
CLb*. (For the
sake of simplicity wedisregard the topological component of n*(-; -).)
Proof. Let T:
p .•• p'
be a morphism of
e.
If [M] En (-;
p),
p
n
we associate a
(p',
n)-manifold, T(M), to M by relabelling all the strata
of M according to the based map T. The correspondence
[M] ..• [T(M)] gives a well defined natural transformation of theories
p p
T*: n*(-;
p) .••
n*(-;
p')
and the functorial properties are clear.
61
If the linked map T:
p .•• p'
is a homotopy equi-
Corollary 3. 2.
the proof of the universal-coefficient theorem.
As pointed out before, if
p
is any based ordered chain complex valence, then T* is an isomorphism.
augmented over G, then the theory n*(-,
p)
can be defined in the same
way. But now the singularities in codimension greater than one are not Proof. This is an easy consequence of the universal-coefficient
necessarily solvable; they give rise to the
E
2
-term of a spectral sequenC11theorem. There is a commutative diagram
60

0- H
(p,
0
(X, A))'"
0
(X, A;
p) -
H
(p,
0
l(X, A)) - 0
°
n n
1
n-
H,(T, 0n(X,
AI)j
j
T. H,
(T,
0n_1 (X,
A)I I
0'" H
(p',
0
(X, A)) -
~i
lX, A;
p) •••
H
(p,
0
l(X, A)) - 0
°
n
j)
1
n-
Obviously the above definition of
n
(X, A;
p)
is independent of the chosen
n
p'
E
e.
Nowwe fix our attention on a particular t.
1.
resolution, called
the canonical resolution of G and written
y.
It is defined as follows:
in which the side-morphisms are isomorphisms, because
T
is a homoto
equivalence. Therefore
T
*
is also an isomorphism.
If G E <tb, a truncate~inked (t.
1. )
resolution of G is an exact
sequence
£
y:r -r, • r-+-G~O
\~2
/0
2'\
~I/a:
FKer£
Lemma 3.3.
!!
lfi:
G ..• G' is a homomorphism of abelian groups,
is a t.
1.
resolution of G,
y'
is the canonical resolution of G';
lfi
extends to a linked map ~:
p -
y'
in a canonical way.
where
r ,
F Ker £, F Ker
lfi
are the free abelian groups on G, Ker E,
°
1
Ker
lfi
respectively
1
a
is the obvious map;
r
= FBI
1 1
B
1
= {(f, w(f)) If E Ker E
C
F Ker E and w(f) is a word expressing
iJ
f in terms of the elements of G
C
r };
1
°
~I(f, w(f))
=
f and
lfil
=
0I~I
r
= FB2; B2
=
{h; w(h), c(h) Ih E Ker
lfi ,
w(h) is a word
2 1
expressing
a
(h), c(h) is a cancellation rule associated to h}. ~ and
2 2
~ are defined similarly
2
y has canonical bases G, BI, B2and a canonical structure in
(h, w(h), c(h)) has (w(h), c(h)) as its structure.
- F
°
-F 1
F
2
(a) M is a submanifold of
oW
(b) FIM
=
f and
F(OW\M)
C
A
(c) Proof. Let
p
= {F , I/>}, y' =
{r',
lfi'
LWe proceed by
M has at most three intrinsic strata (labelled by elements p p p P
of BO, BI, B
2).
induction on p. Write (~) = (~o' ~I' ~/ For p
=
0, put ~o(bo)=I/>E(bo)
° ° ~
p -
W is called a bordism. Define -(M, f)
=
(-M, f). Two singular
(p,
n)- for each b EB. Inductively, let E B. ~hen I/>p_1I/>p(bP)EKerl/>~_l
1 (M f) (M f) b d t Ofth dO " "t " and therefore it determines a basis element, b , in F Ker I/>p'1; b'P has
cyc es , , , are or an 1 e 1Sl01l1umon -
(M U M
1
f IUf ) ~ d
2"
(X A) B d"
° °
1 1 t" a canonical word w(b'P) and cancellation rule c(b'P) induced from those
1 - 2' 1 2
or s 111 , . or Ism IS an equ1va ence re a lOn .
p __
of
0-
through the map (I/>
lfi)
Therefore the assignment
in the set of singular
(p,
n)-cycles of (X, A). Denote the bordism class of p-l' p-£'
- tfl-
(b'P w(b'P) c(b'P)) defines
lfi
with the required properties.
(M, f) by [M, f] and the set of all such bordism classes by On(X, A;
p), , "
p
An abelian-group structure is given in Un(X, A;
p)
by disjoint union.
I
satisfying conditions (a), (b), (c), (d) in the definitions of structured
resolution given in Section
2.
A linked map between t.
1.
resolutions is
defined as in the non-truncated case and there is a category,
e,
whose
objects are t.
1.
resolutions and whose morphisms are linked maps.
In the following, for each
p
E~ and each topological pair (X, A),
we construct a graded abelian group
{n*(x,
A;
p)}
which is a functor
on
CP
x ())
(G
= category of topological pairs). Fix a
p'
E
e
obtained
from
p
by choosing a based kernel of
lfi.
A singular
(P,
n)-cycle in
2
(X, A) is a pair (M, f) consisting of a
(p',
n)-manifold M and a map
f : (M,
oM) -
(X, A) such that M has at most two intrinsic strata labelled hOh
w
1C
by elements of B
°
and BI. A singular
(p,
n)-cycle (M, f) is said to
bord if there exists a
(p',
n+l)-manifold Wand a map F : W-X for
which
62
63

Theorem 3.6.
t
G: n*(X, A;
PG) ....•
n*(X, A; G) is a natural
equivalence of functors.
Q/ is an epimorphism: it follows from commutativity and the
fact that t is epi.
1, 3
Q/ is a monomorphism: let Mnbe a (singular) G-manifold such
that Q/(Mn)_
¢
in n (X, A;
p).
Then Mndetermines an element
n
3
[M]
E
n (X, A;
P )
such that t [M]
=
O.
Since t is a mono-
P
2
n
2 2,3
P
22,3p
morphism, we deduce that there is a
p
bordism W: Mn~
ro.
Finally
2
we observe that
p
'+
y
is a linked embedding of resolutions and there-
2
fore W provides the required bordism of Mnto zero in 0 (X, A; G).
n
The proof of the theorem is now complete.
We are now able to state the theorem asserting the possibility of
E
making bordism with coefficients in a short linked resolution
p ....•
G
depend functorially on G.
Theorem 3. 7. (a) There exists a (graded) functor
U*(X, A; G) :
<.P
x Cib'" Cib* which associates to each pair (X, A; G)
the graded abelian group n*(X, A;
PG)
where
PG
is a fixed linked
l,presentation of G; to every morphism (f,
cJ» :
(X, A; G) ....•(X', A', G')
. -1
the graded homomorphIsm (f*, tG,
cJ>*t
G) :
n*(X, A;
PG) .•
n*(X, A;
PG,).
(b) Functors corresponding to different choices of
PG
are
In
,aturally equivalent.
n*(x, A;
p)
gives a functor
<.P
x
e .•
Cib*,
(a)
Lemma 3.4.
In view of the previous corollary we shall write n* (X, A; G)
instead of O(X, A;
y)
and
cJ>*
instead of ¢*. A
(y,
n)-cycle [bordism]
will also be called a (G, n)-manifold [(G, n)-bordism].
A linked resolution of an abelian group G
is said to be p-canonical (1:S p :5 3) if
E
F 1" ....• G .•
0
is the canonical resolution of G of
p-
length p - 1, i. e. F.
=
r.,
O:s
i :5P - 1, and the morphisms
cJ>.
are
1 1 1
the same as in the definition of
y.
Proof. Functoriality on
<.P
is obvious,
If
T:p ....•p'
is a morphism of
e
and [M]
d~
(-;p),
let T(M) be
p
n
the
(p',
n)-manifold associated to M by relrcbelling each stratum according Proof. Let
Pi
be an i-canonical resolution for i
=
2, 3. Then
to T. The correspondence [M] .• [T(M)
1
gives the required natural trans- there is a commutative diagram
- - P P
formation T* : n*(-;
p) ....•
n*(-;
p').
tGt
12
-
,
:~~~F:::or~i:'0:;
;1
::::i~~ndorWx
"b - db.,
O.<X,
S
0.0<,~~,
-7,,~;
p,)
To each G
E
Gb assign n* (X, A;
y)
where
y
is the canonical Q/ ~ n*(X, A;
p )
resolution of G; to each morphism
cJ>:
G ....•G',
cJ>
E
Cib, assign the homo-
3
morphism ~* : n*(X, A;
y) ....•
n*(X, A;
y'),
where
¢
is the canonical where t .. and Q/ are the natural transformations obtained in the usual
1,
J
extension of
cJ>
described in Lemma 3. 3 and ¢* is the induced homo- way by relabelling the cycles according to the canonicallifting's of
morphism described in Lemma 3.4. id : G .• G. By Corollary 3.2, t. . is an isomorphism (1:5 i
<
j
:5 3).
1,
J
Therefore, in order to prove the theorem, we only need to show that Q/
is an isomorphism.
(b) Fp+1
=
Fp+2
=
O.
Let G
E
Cib and
PG
E
e
a short linked resolution of G, (i. e.
F
2
=
F
3
=
0).
Then we have the homology theory {n*(X, A;
PG)'
a]
defined in Section 1. We also have the graded functor 0* (X,A;G) :
<.P •••.•
Cib* constructed at the beginning of this section. It follows from
Lemma 3. 3 that there is a canonical extension of the id: G •.•G to a
linked map id:
PG....•
y.
The latter induces a natural transformation of
functors tG: n*(X, A;
PG) :
O*(X, A; G) obtained by relabelling the
PG-manifolds according to id. The next theorem is the main step
towards functoriality.
64
65

The result follows immediately front Theorem 3.6 and Corollary
3.5.
With the notations of the theorem, we define n*(-; G), the p.l.
oriented _bordism with coefficient group G, by n*(X; G)
=
n*(X;
PG).
Corollary 3. 8. For every pair X, A, every n
2':
0 and every
abelian group G, there is a short exact seguence
o
-+G
@
n (X, A) -+n (X, A, G) -+Tor(G, n l(X, A)) -+0
n n n-
which is natural in (X, A) and in G.
We are now able to say something about the splitting of the uni-
versal coefficient sequence associated to n*(-; G). Precisely, since
the sequence is natural on the category ab, Hilton [2; Theorem 3.2],
gives us the following
Corollary 3. 9. For every abelian group G, the universal-
co~fic~nt sequence of nn (-; G) is pure.
From algebra we deduce:
Corollary 3.10. For every pair (X, A), abelian group G and
integer n
2':
0 such that Tor(nn_1 (X, A), G) is adirect sum of cyclic
groups, the universal coefficient seq~ence
0-+ n (X, A)
®
G -+ n (X, A, G) -+Tor(rl l(X, A), G) -+ 0
n n ~
splits.
The class of examples of splitting considered by the previous
corollary is quite vast, because it includes the following cases:
(a) any G finitely generated
(b) any G such that its torsion subgroup has finite exponent
(c) any n 1(X, A) such that its torsion subgroup has finite
n-
exponent.
Remark 3.11. As we have pointed out earlier, the definition of
(p,
n)-manifold makes sense in the case of
P
being any linked chain
complex (not necessarily a resolution) and there is an associated bordism
66
theory n*(-;
pl.
Some of the facts about morphisms, that we have
established in this section, hold in the case of chain complexes. In
particular, if
T:
p
-+
p'
is a linked chain map,
P
is a linked chain
complex and
p'
is a linked resolution, then the proof of Proposition 3. 1
applies to give an associated morphism
T*: rl*(-; p)
-+n*(-;
p').
The above treatment of functoriality can be summarized as
follows. For every abelian group G, two functors
<P
-+ G.b* have been
set up, namely n*(X, A;
PG)
and O*(X, A; G). They have different
features: the former is readily seen to be a generalized homology theory;
while the latter is natural on the category of abelian groups. Theorem 3. 6
establishes a natural equivalence tGbetween the two functors, which
~roves at the same time that n*(X, A;
PG)
is natural on ab and that
1l*(X,A; G) is a homology theory.
In the following we may use whichever of the equivalent functors
1l*(X,A;
PG),
n*(X, A;
Pi)'
S1(X,A; G) is more appropriate to the
context (i
=
1, 2;
P.
=
any i-canonical resolution of G).
1
4.
PRODUCTS
If G, G' are abelian groups, let
P
be a linked resolution
</I E
O-+F -+ F -+ G-+O
1 0
with
B
O
=
G
=
{gl' g2' •••
B
1
=
{r
l'
r
2' ••. }
and
p'
defined similarly. Then
p
@
p'
is the augmented chain complex
IF",
</I"}
</I" </I"
2 1 E"
0-+ F
@
F' .•• F
@
F' tBF
@
F' .•• F
@
F' -+ G
@
G' .••0
11011000
where
1J!"(r
@
r')
=
</I(r)
@
r' - r
@
</I'(r')
2
1J!"(g
@
r')
=
g
@
</I'(r')
1
</I"(r
@
g')
=
</I(r)
@
g'
1
E"
=
E
@
E'
67

g'
2
r
I8l
g'
3
r
I8l
g'
2
Picture of a
neighbourhood of
SM x (M' - SM')
in
M
x
M'
SM'
g
I8l
g'
3 3
~,
g7;t"
g; :'" g~'
2 3
r®
g'
3
g
®
g' -
r
I8l
g'
1 1@V:,\~ 1
r
SM
gI8l
r'
1
g@r'
3
basic link of a
point of SM x SM'
in
M
x
M'
Fig.
12
(M - 8M) x (M' - SM')
SM x (M' - SM')
U
(M - SM) x SM'
SM x SM'
p
@
p'
is based by means of B\ B,i (i
=
0,
1);
it is structured in
dimension one by the structures of
p
and
p';
in dimension two we
assign to r @r'
E
B,2 the word w(r
®
r')
=
w(r) @r' - r @w(r'). For
now we do not fix a cancellation rule.
Let M be a
(p,
i)-manifold with singularities SM; M' a
(p',
j)-manifold with singularities SM'. Form the cartesian product
M x M'. It has three intrinsic strata given
by
We show that M x M', with such additional structure, is a
I
I
(p
®
p',
i
+
j)-manifold. The first and the second stratum are easily
I..•.1.1'.',
seen to be
p
@p'-manifolds of the appropriate dimensions and we are
going to examine the third stratum. For simplicity assume SM, SM' ,
I
constantly labelled by r, r' respectively, so that SM x SM' is labell1
by r
®
r'
E:
B,,2. The basic link of SM x SM' in M x M' is topologiq
the join L"
=
L(r, p)*L(r',
p').
L" with the structure induced by M
11'
is a
(p
@
p',
I)-manifold because M x M' - SMx SM' is a
(p ®p')- \;
d
manifold and there is a product structure around SM x SM'. The zerof
i
dimensional stratum of L" is isomorphic to L(r,
p)
@r' Ur@-L(r',
pi
where @ is meant to act on the labels. But this represents the word :
,
~
w(r
®
r'). Thus L" is a
(p
@
p',
I)-manifold in which the zero-
t2~.
dimensional stratum represents w(r @r') and the I-dimensional stratI
is a union of disks. Therefore L" gives a unique cancellation rule to ~
I
assigned to r @ r' in order to' have L"
=
L(r @r',
p
@
p'). ,
M
x
M' ;., then a
(p "p',
i+j)-manilold. See Flg,. 12 and 13.\
J
'.}
On each stratum we can put labels via the tensor product, i. e. if V is
a component labelled by x : V' labelled by x', V x V' is labelled by
x
@
x'.
Fig. 13
68
69

We can now define a homomorphism
5. THE BOCKSTEIN SEQUENCE
x : n (-'
p) ®
n (-'
p') -
n (-'
p®p')
p, p'
*' *' * '
by
Theorem 5.
1.
On the category of short exact sequences of
abelian groups
'" ljJ
o -
G' - G - G" - 0
there is a natural connecting homomorphism
-
-
(3 :
n*(-;
G") -
n*(-;
G')
x
p, p'
is of degree zero.
Let
p
be a 3-canonical resolution of G
®
G'. Then, by the
3 _
proof of 3. 3, there exists a canonical lifting id:
p
®
p' -
P
of
3
id: G
®
G' -G
®
G'. Therefore we can define a cross--,pc:r:...;o:..:d::..:u::..:cc::.t
homomorphism'
f
dId t
11
o
egree - an a na ura ong exact sequence
X
GG' : n*(-; G)
®
n*(-; G') - n*(-; G
®
G')
,
Remark 4.
1.
If an abelian group G is also endowed with a multi
plication that makes it into a ring, then we have a product homomorphism:
Proof. For the sake of clarity of exposition we prove the theorem
underthe assumptions: G'
C
G and (X, A)
=
(point,
¢).
Realize the exact sequence of abelian groups by the (not neces-
sarily exact) sequence of canonical resolutions and linked maps
r'
'+
r
-
r"
2
2
2
~~ ~
r'
e-.
r
-
r"
11 1
~~
*
r'
'+
r-r"
000
l
~
l
o -
G'
'+
G
-
G"
-
0
(I)
Definition of (3
Let Mnbe a G"-manifold. Suppose that the singularities of M
ave only one connected component V
®
r", with r"
=
g~
+ ... +
gt a
relation in G". We relabel V by an element of G' as follows. Choose
1, ••• ,
gt
E
G such that ljJ(g.)
=
g? and g.
=
g. ~ g?
=
g~'. Then
1 1 1
J
1
J
(gl
+ ... +
gt)
=
g;
+ '" +
gi'
=
0 in G". Therefore by exactness
,=
L.g. is an element of G'. We relabel V by g' and get a
1 1
G',
n-l)- manifold V
Q\)
g'.
n*(-
;lP~~G')
t
1, 3
n*(-;
P3)
.'
x
n*(-;
p)
®
n*(-;
p')
G1.P~ __ ~
x
p,p'
by the composition
-
where
PG®G'
is a linked presentation of G
®
G' and id* is the usual
relabelling map (as in the proof of 3. 6).
where x G G is the cross-product and m: G
®
G - G is given by:
,
m(g
Q\)
g')
=
gg'. The homomorphism
Il
makes n*(point; G) into a
ring and if (X, A) is a pair, n*(X, A; G) can be given a structure of
graded module over the ring n*(point; G) in the usual way.
70
71

-
-
!3 :
n (-'
G") -
n (-'
G')
n ' n-l '
a 'relation amongst relations' of G". We label Tn-1by r'. If (3(W)
denotes SW with the relabelling described above, then !3(W)provides
the required G'-bordism !3(M)-
0.
Nowwe are entitled to define
g'
g'
g'
2
Fig. 14
g V®r g
1
I
3
Ck;J
g3
2
Now suppose g], ... , gt is another lifting of g~, ... , gi' as
above, giving a (G', n-l)-manifold V
®g'.
We show that V
®
g' and
V ®g' are bordant as (G', n-l)-manifolds. Relabel M\V by changing
g~' into g~
=
g. - g.
EO
G', i
=
1, ... , 1. The sum r' = (g' + ... + gt')'
1 1 1 1 1
(g' - g') is a relation in G'. Therefore take a labelled copy V
®
r'
of V and form the polyhedron W
=
(V
®
r') x L(r', G') UM, where
L(r', G') is the link generated by r' in G' and the union is taken along·by
the common part V x cone(g~ + ...
+
gP (see Fig. 14). W is a
(G', n)-manifold and provides the required bordism between V
®
g' and
V
®g'.
(2) Exactness at 0 (-; G)
n
(a) lJI*lJl*
=
o.
If M' is a (G', n)-manifold, then M"=lJIf/>(M')
is a
(0,
n)-manifold. Therefore M" -
0
by the proof of the universal-
coefficient theorem.
(b) Ker
lJI*
C
1m lJl*. Let Mnbe a G-manifold and W" a
(G", n+l)-manifold with
aw"
= lJI(M). We show how to modify W" in
order to get a G-bordism between Mnand a G'-manifold M,n.
Relabel each component of the (n+I)-stratum of W" by elements
of G, obtained from the G"-labels through a lifting G
t
G" such that
..•.
-
-
(a) The relabelled n-stratum of
aw"
coincides with the n-
stratum of Mn.
(b) If two components are labelled by the same element of G",
If the singular set, S(M), of M has more than one component, the corresponding liftings coincide.
the relabelling construction can be performed componentwise and one Let V be a component of the n-stratum of W" and g , ... , g
V 1 V
gets a (G', n-l)-manifold, !3(M), whose bordism class is independent of the new G-labels around V; g'
= ~
g. is an element of G'. Attach a
the various choices. i=l
1
Next we show that !3(M) depends only on the bordism class of new sheet (V x I)
®
g' to V iff g' '"
0
and label V by the G-relation
M, i. e. if M
<2"
(0,
then !3(M)~'
0.
Let W be a (G", n+1)-manifold
wi
r = Ligi- g' (see Fig. 15). Now let V
®
r,
V
®
r,
V
®
r, ..,
be the
aw
=
M. If V
C
W\f3W
is a component labelled by g"
EO
G", choose components of the relabelled n-stratum merging into a component T of
g
EO
G such that lJI(g)= g" and relabel V by g. Let Tn be a com- the (n-l)-stratum. The corresponding new sheets which have been in-
serted, namely (V x I)
®
g',
(V
x I)
®g',
(V
®
I) x =g', •.. , are, by
ponent of the n-stratum of W. The sum in G of the new labels on the
sheets coming into Tn is an element g' of G'. We relabel T by g'. construction, such that r' = g' + g' +
g'
+ ... is a relation in G'.
Finally, if Tn-l is a component of the (n-l)-stratum of Wand Therefore we can glue them to one another along a new n-dimensional
Tn
®
g', ... , Tn
®
g' are the sheets merging into Tn-I, then sheet (T x I)
®
r'. The resulting polyhedron W provides the required
lIS S
G-bordism.
r' = ~ g: is a relation in G', because Tn-l was originally labelled by
i=ll
72
73

Fig. 15
(3) Exactness at 0n(-; G")
(a)
{3lJ1*
= O. Let Mnbe a G-manifold. According to the
definition of (3, we have
(3lJ1(M)
='---'V ® 0, where V varies over the
V
set of components of S(M). But V ® 0 -
¢
by a trivial G'-bordism.
(b) Ker {3
C
1m
lJI*.
Let M"n be a G"-manifold. For the
sake of simplicity let us assume that the singularities of M" have only
one component V ® r"; r"
=
L. g~. Then (3(M")= V ® g', where
I I
g'::;: L.g.,
lJIg.
=
g~. By assumption there is a G'-bordism W' : (3(M")-~,
I I I I
We construct a G-manifold Mnas follows.
(i) Since {3(M") has no singularities we can assume that W' has
singularities in codimension one at most, because otherwise we solve
the codimension-two stratum as in the proof of the universal-coefficient
theorem.
74
(ii) In M" we replace each g~' by g. and V ® r" by V ® r, where
I I
r
=
g' - L.g ..
I I
(iii) We attach W' to the relabelled M" identifying
aw'
with V®r.
It is readily checked that the resulting labelled polyhedron Mn
G"
is a (closed) G-manifold such that
lJI(M) -
M".
See
Fig. 16
g"
C>C)
g~ g~
Fig. 16
(4) Exactness at 0n(-; G')
n+1
(a)
q,*f3
=
O. Let M" be a G"-manifold with connected
singularities V ® r", r" = L.g~'. Then f3(M")= V ® g' where
I I
g' = L.g. and
lJIg.
= g~. We construct a G-bordism W: (3(M")-
yf
as
I I I I
follows.
(i) We relabel M by changing each g~ into g. and r" into
I I
r
=
L
.g. - g'.
I I
(ii) We attach a new sheet (V x I) ® g' to the relabelled M" along
the singularities V ® r. See Fig. 17
Fig. 17
75

6. BORDISM WITH COEFFICIENTS IN AN R-MODULE
The proof of exactness is now complete and naturality is clear.
In order to define coefficients in an R-module we need the following
additivity lemma.
.•• 0
.•• 0
---
~
Proof. Consider the chain map
1/1
=
(II
+ {3-
«(II
+ (3) where
--
-
(II,
{3,
(II
+ {3are the canonical liftings of
(II,
{3,
(II
+
f3.
If [M]
dl (-;
G),
- __ ~ _ n
put
I/I(M)
=
(II(M)
+
f3(M) - «(II
+
(3)(M).
Then
I/I(M)
is a (G', n)-manifold
and we only need to prove that ~(M) -
¢
in
U
n(-; G'). But ~ is a lifting
of the zero map 0: G'" G', Therefore there exists a chain homotopy
Lemma 6. 1.
!!
(II,
{3: G'" G' are abelian-group homomorphisms,
then
«(II
+ (3)*
=
(11*
+
f3* :
U*(-;
G)'"
U*(-;
G'),
-
D:
1/1""
O.
q,2 q,l
e;
r
...
r
...
r
...
G
2 / 1 /
a
1°
l/~;
h{" 1~"
r'
...
r'
...
r'
...
G'
2 1
a
so that
1/1
=
D
q,
+
f/I'D
1
a
1 2 1
1/1
=C/>'D.
o
1 0
By definition the singularities of
1/1
(M) are given by
1/1
(SM). Consider
the induced diagram
(b) Ker
q,*
c
1m p, Let M,n be a G'-manifold and W: M,n -
¢
a G-bordism, From W we get a G"-manifold W" of dimension (n + 1)
as follows,
(i) We remove from W all the strata which are labelled by elements
of G' or by relations or by 'relations amongst relations',
(ii) We relabel the remaining strata according to the map
1/1.
The
resulting object W" is a (G", n+I)-manifold with singularities in co-
dimension 52 and it is closed because
aw
=
M' has been removed in
step (i),
(iii) We get W" by re-solving the codimension-two singularities of
W"
up to a bordism which has singularities in codimension
53.
Nowwe show that (3(W") is G'-bordant to M', Let Q" be the
singular part of the bordism used in (iii). Q" has at most two sheets;
the non-singular one is labelled by relations in G" and the singular one
is labelled by 'relations amongst relations'; (3(Q") (constructed as in
the proof that the Bockstein is well defined) realizes a G'-bordism between
(3(W") and {3(W"), Therefore we only need to provide a G'-bordism N'
between (3(W") and the original M', To this purpose we reconsider the
G-bordism Wand remove from it all the top dimensional strata which
are not labelled by elements of G', The resulting object W
o
is not a
G'-manifold in general. We show how to make W into the required
a
G'-bordism by inserting new sheets.
(i) Let V
®
r be a component of the n-dimensional stratum of W,
with r
=
g' + . " + g' + g +, .. + g (g~
E
G'; g.
E
G - G'). Then
1
p
1
q
1
J
V
®
g'
C
(3(W") where g'
=
g +." + g, Therefore we attach a sheet
1
q
(V x I)
®
g' to W along V and change the label r into
a
r'
=
g' +", + g' + g', which is now a relation in G',
] p --
(ii) Let V
®
r,
V
x
r,
V®
r, '"
be components merging into a
component 'I'
®;
of the (n-l)-stratum, where;
=
r +
r
+
r
+ '" is
a relation amongst relations in G. The corresponding new sheets which
have been inserted, namely (V x I)
®
g',
(V
x 1)
®g',
(V
x I)
®
g', ,."
are, by construction, such that r'
=
g' +
g'
+
g'
+ , " is a relation in
G', Therefore we can glue them together along the n-dimensional sheet
('I' x I)
®
r', The resulting polyhedron provides the required G'-bordism
N' :
(3(W") -
M',
76 77

where
a"
is the R-module structure of G.
The pair {n*(-; G),
a}
is 'bordism with coefficient~}n the R-
module G'. The structure
a
will be dropped from the notations.
If f : G'" G' is an R-homomorphism, then for every r ER we
have commutative diagrams
2
E
=
Tor (n (-'R) G)
=}
n (-'G)
p,q p q' , p *'
This completes the discussion of the case of R- modules as co-
efficients. In later chapters we shall only deal with abelian groups; but
it is understood that everything we say continues to work in the category
of R- modules.
Hence n*(-; G) is a functor on the category of R-modules and R-homo-
morphism. From the naturality of the Bockstein sequence for abelian
groups, it follows that there is a functorial Bockstein sequence in the
category of R-modules. Summing up, we have the following:
(a) n*(-; G) is a functor on the category of R-modules
(b) n*(-; G) is additive
(c) For every short exact sequence of R-modules, there is an
associated functorial Bockstein sequence.
Properties (a), (b), (c) form the hypothesis of Dold's Universal-
coefficient theorem [1].
Therefore we deduce that there is a spectral sequence running
f*
n*(-; G) ---.- n*(-; G')
a(r)! func~:riality! a(r)
n*(-; G)
----1._
n*(-; G')
G •• G'
V'
=
ljI(V)
=
(1 ®'/>')(1 ® D )(V)
o
-
-
ljI(SM)=(l®ljI
)(SM)=(l®D
)0
(l®CP)(SM)+(l®lJl')
0
(l®D )(SM).
1 0 1
so that V' may be borded to
9f
by a (G', n+1)-manifold with singularities
given by (1 ® D )(V).
o
We now turn to the main object of this section, i. e. putting co-
efficients in an R-module. In the following R will be a commutative
ring with unit.
If G is an R-module, let n*(-; G) be bordism with coefficients
in the underlying abelian group G; n*(-; G) has a natural R-module
structure. In fact, we must exhibit a ring homomorphism
a:
R ..•HomZ(n*(-; G), n*(-; G)). The above additivity lemma, together
with functoriality, tells us that there is a ring homomorphism
But (1 ®lJl)(SM)-
9f
because [SM]= Ker(l
®
CP).
Therefore
_ 1
ljI(SM)
=
(1
®
lJl')(W'),where W'
=
(1
®
D )(SM). By the proof of the
1
universal-coefficient theorem this is sufficient to ensure that the singu-
- -
larities
ljI(SM)
of
ljI(M)
can be resolved by a bordism of (G', n)-
manifolds. So, using the homotopy D, we have eliminated the singular
stratum of
ljI(M).
Let V' be the resulting (G', n)-manifold. If V' is
o
a component labelled by g'
E
G', then there exists g
E
G such that
_ 0 0
ljI
(g )
=
g', because the process of resolving the singularities in
ljI(M)
o
0 0
does not change the labelling of the top dimensional stratum. Therefore
the element of n (-) ®
r'
represented by V' is the image of some
n
0_
[V]
E
n (-)
®
r
through 1 ®
ljI.
Then again we have
n
0 0
-
where 1 ® D is now a homotopy of 1 ®
ljI
to zero. As we know, SM
represents an element in n 1(-) ®
r.
Sowe have
n-
1
REFERENCES
FOR CHAPTER ill
defined by
a'(f)
=
f*. Therefore we can define
a
by the composition
a
R ..• HO~(n*(-; G), n*(-; G))
a~ la'
Horn (G, G)
Z
[1]
[2]
A. Dold. Universelle Koeffizienten. Math. Zeitschrift, 80
(1962/3).
P.
J.
Hilton. Putting coefficients into a cohomology theory.
Konikl. Nederl. Akademie van Weterschappen (Amsterdam),
Proceedings, Series A, 73 No. 3 and Indag. Math. 30 No.3,
(1970), 196-216.
78 79

[3]
P.
J.
Hilton and A. Deleanu. On the splitting of universal co-
efficient sequences. Aarhus Univ., Algebraic topology Vol. I,
(1970), 180-201.
[4]
J.
Morgan and P. Sullivan. The transversality characteristic
class and linking cycles in surgery theory. Ann. of Math. 99
(1974), 463-544.
80
IV·Geometric theories
In this chapter we extend the notion of a geometric homology and
cohomology (mock bundle) theory by allowing
(1) singularities
(2) labellings
(3) restrictions on normal bundles.
The final notion of a 'geometric theory' is in fact sufficiently
general to include all theories (this being the main result of Chapter Vn).
A further extension, to equivariant theories, will be covered in Chapter V.
In the present chapter, we also deal with coefficients in an arbi-
trary geometric theory. A geometric theory with coefficients is itself
an example of a geometric theory and it is thus possible to introduce co-
efficients repeatedly!
The chapter is organised as follows. In
§
1 we extend the treat-
ment of coefficients in the last chapter to cover oriented mock bundles
and in
§ §
2 and 3 we deal with singularities and restrictions on the normal
bundle. In §§4 and 5 we give interesting examples of geometric theories,
including Sullivan's description of K-theory [11] and some theories which
represent (ordinary) Z -homology. Finally
§
6 deals with coefficients
p
in the general theory.
1. COBORDISMWITHCOEFFICIENTS
We now combine Chapters n and III to give a geometric description
of cobordism with coefficients. It is first necessary to introduce oriented
mock bundles (the theory dual to oriented bordism). We give here the
simplest definition of orientation, an alternative definition will be given
in
§2.
Suppose Mn, Vn-1are oriented manifolds with V
C
aM. Then
we define the incidence number £(V, M)
=
±1 by comparing the orienta-
81

tion of V with that induced on V from M (the induced orientation of
aM
is defined by taking the inward normal last);
E(V,
M)
=
+1 if these
orientations agree and -1 if they disagree. An priented cell complex K
is a cell complex in which each cell is oriented and then we have the
incidence number
E( T, a)
defined for
Tn-l <an E
K.
An oriented mock bundle is a mock bundle
UK
in which each
block is oriented, K is oriented and such that, for each
Tn-l
<
an
E
K,
we have
E(~(T), ~(a))
=
E(T, a).
We leave the reader to check that the
theory of oriented mock bundles enjoys all the properties of the unoriented
theory in Chapter II (the Thom isomorphism theorem holds for oriented
bundles and Poincare duality for oriented manifolds) - more general argu-
ments will in fact be given in §2. This theory will be denoted n*(,) and
the dual bordism theory n*(,).
Now let G be an Abelian group and
p
a structured resolution of
G. We define the mock bundle theory n*(,;
p)
by using p-manifolds in
place of ordinary manifolds. More precisely, a
(p,
q)-mock bundle
UK
is a polyhedron E(~) with projection p: E(~)
-+
K
such that, for
each
(Ji
E
K, p-l(a) is a
(p,
q+i)-manifold with boundary p-l(&), called
the block over
a
and denoted
~(a),
and such that
E(~(T), ~(a))
=
E(T, a)
for each
T<a
E
K. Note,
E(V,
M) is defined for p-manifolds
V
n- \
Mnonly when either V or -V
c
aM
as p-manifolds (i. e. the inclusion
respects the labellings, orientations and extra structure), then
E(V,
M)
=
+1 in the first case and -1 in the second.
-V
denotes the
p-manifold obtained from V by reversing aU the orientations.
It follows from the arguments in Chapters I and II that n*(, ;
p)
is a COhomologytheory, dual to the theory n*(,;
p)
defined in Chapter
1lI, and from the arguments in III §4that n*(p, Q; -) determines a
functor on the category of abelian groups.
We will leave most of the details to the reader and make some
remarks about some of the more delicate situations:
Remarks 1. 1. 1. If
~;K
is a
(p,
q)-mock bundle and
I
K
I
is
an (oriented) n-manifold, then E(~) is a
(p,
n+q)-manifold. The proof
of this is identical to the proof in Chapter il - the required extra struc-
ture all comes automatically!
82
2. In order to prove Poincare duality, one needs Cohen's
transversality theorem in its full generality, i. e. if f:
J
-+
K is sim-
plicial, then f-l(A) is collared in f-l(A*) for each A
E
K. Here A*
is the dual cone of A in K with base A. From this theorem it follows
that, if f: E
-+
K is a simplicial map, E is a
(p,
n+q)-manifold and
I
K'
is an n-manifold, then the inverse image of a dual cell in
K
cuts
the singularities of E transversally, so that f: E
-+
K can be made
into the projection of a
(p,
q)-mock bundle.
3. A discussion completely analogous to that of ill §4can be
carried out. In particular there are functors n*(,; G) natural on the
category of abelian groups and there is a universal coefficient sequence
also natural on the category of abelian groups.
4. If
¢:
G
®
G'
-+
G" is a pairing, the cup product
n
q( ; G)
®
nr( ; G')
-+
nq+r( ; G") and the cap product
n
q( ; G)
®
nr( ; G') •.•nq+r( ; G") are defined using the usual pull-back
construction and the cross product defined in Chapter
m.
2.
RESTRICTIONS ON NORMAL BUNDLES
In this section we consider geometric (co)-homology theories
which can be obtained from pi (co)-bordism by restricting the normal
bundles of the manifolds considered. We sketch the case of cobordism.
Details for the bordism case may be found in
p3;
Chapter
ill.
p
Let E(~)
-+
K be a mock bundle projection, then we can choose
00
an embedding i: E(~)
-+
K x R so that p
=
11
0
i, we then have a stable
1
normal block bundle
V~IE(~)
on E(~) in K x
ROO.
There is a classifying
bundle map
~
TJ~
E(v ~)
~
E(y)
I)
U
E(~)
••
BPt
TJ~
83

85
3. SINGULARITIES
Our treatment of singularities is similar to that worked out by
,Cookeand Sullivan (unpublished) or to be found in Stone [9].
Suppose we are given a class £ of (n-l)-polyhedra(closedunderpl
n
isomorphism). Then a closed £ -manifold is a polyhedron M each of
n
whoselinks lies in £. A theory of manifolds-with-singularity consists
n
ofa class oC for each n
=
0,
1, .. ,
which satisfies:
n
each member of oC is a closed oC I-manifold
n n-
SoC
1
c
oC (1.e. the suspension of an (n-l)-link is an n-link)
n- n
CoC 1 n £
=
0
(1.e. the cone on an (n-l)-link is never an
n- n
(2. 1)
m
BPL
XxX
I
f
x
f
-
-
BPL x BPL
,..
E(~)
84
1.
2.
3.
-link).
where
E9
is the map given by Whitney sum. Using diagram (2. 1) external Then an oC-manifoldwith boundary is a polyhedron whose links lie
products can be defined by
q~x71
=
m
0
(q~ x q1))'
Similarly cap products ither in oCnor CoCn_l' Then the boundary consists of points whose
are defined with the corresponding bordism theory (maps of (X, f)-mani- inks lie in the latter class, and is itself a closed oC I-manifold. More-
n-
folds into the space) and the proof of Poincare duality (for (X, f)-mani- ver the boundary is locally collared (since its links are cones) and
folds) needs little change. The proof of the Thorn isomorphism theorem ence collared [8; 2.25].
for bundles with stable lifts in X can also be readily modified. Notice that axiom
3
is necessary to ensure that the boundary is
~
ell-defined. Axiom 2 ensures that if M is an oC I-manifold then
E 1 2 2 1 O' e ted theory X
=
BSPL and f is the n-
xamp es ., .
n
n .• x I is an .£n-manifold with boundary. Axiom 1 implies that a regular
natural map. This theory has products. See also the alternative des-
f
..
eighbourhood of a polyhedron in an £n-manifold is itself an oCn-manifold
cription given in §l. ..ith boundary.
2. Smooth cobordism. X has the homotopy type of BO and At this point we can remark that a manifold with singularities has
f : X'" BPL is defined using
PD
as in
[7;
§O]. This again is a theory III the geometry of an ordinary manifold which was used in setting up
with products .. ordism and cobordism (mock bundles) and we get homology and co-
f BSPL d.omology theories Tt,(,), T£*( ,). Moreover the proofs of the Poincare
3. Pl
spin cobordism. X is the double cover
0
an
f "-'
uality and Thorn isomorphism theorems are unaltered. Note however,
f is the covering map. Again we have products •. .ee below, that products are not defined in general (but cap product with
I
I
I
The theory of (X, f)-mock bundles is set up in exactly the same
way as the theory of bundles. In order to have products one needs in
addition a commutative diagram
,
j
where y/BPL is the classifying bundle for stable block bundles.
I
4. Stable cohomotopy. X is contractible. Again we have
Nowsuppose that we have a space X and a fibration f: X'" BPi. ,products. Poincare duality holds for lI-manifolds.
Then an (X, f)-mock bundle is a mock bundle ~ togeAtherwith a stable
I
5. Labelling. Let S be a discrete set and let X
=
BPL x S
normal block bundle
v
f
a classifying bundle map
(fJ
f
71~) and a lift ,and f the projecti~ Then a connected (X, f)-manifold is just a mani-
of
fJ ~
in X: IIOldlabelled by an element of S. Any function S x S ..• S gives this
theory products. See also the remarks at the end of the next section.

87
the point. (§4 contains details of killing. )
, To combine £-theory with the restriction on the normal bundle
:,'Ofthe last section, it is necessary to use the notion of 'normal block
lbundlesystem' as in Stone [9]. The resulting theories then enjoy all
jtheusual properties - the class of bundles and manifolds for which a
!theory has Thorn and Poincare isomorphisms depends on the stable
jrestrictions imposed on the normal bundles. Rather than attempt a
.j1rormalanalysis of this general setting, we will give several examples
,insubsequent sections, which should make the general properties of
lthese theories clear. We already have the examples, in Chapter Iil, of
coefficients (all the structure of a manifold with coefficients
p
is in-
is the class of (n-l)-spheres. As men-
The set of basic links
4.
86
2. Basic links are
£0
=
{p'),
£1
=
{xix ~
SO
sq-2
*
(n points)
J,
q
2:
1.
Examples 3.
1. 1.
£
n
tioned above, this is ordinary bordism theory.
here is {
P'
J.
eludedin 'restriction on the normal block bundle system') and in
Chapter VII, we will give a family of examples, generated by the killing
process of §4, below, which include all homology theories.
A subset ill of £
=
U
£n is basic if no link in ill is a suspension Finally we remark that we have now arrived at the general notion
and each link in £ is isomorphic to a suspension of a link in ill. ofa geometric theory, since 'labelling' is included in 'restriction on
Jnormalbundle', see Example 2.2(5).
l
j
11,
KILLING AND K-THEORY
I
In this section we give the general description of 'killing' an
I
E
£ and (n points)
E
£. Thus
I
o
1
q-l lelementof a theory and apply it to give a geometric description of
or (n points)
J,
£
=
{xix ~
S or
1
q fonnected K-theory at oddprimes due essentially to Sullivan [ll]. See
.. ~lsoBaas
[15].
Killing is defined in the following generality:
This theory is 'twisted Z -manifolds'. A mamfold m the theory
1
n n-l
.!
1. U and V are geometric theories.
is either locally an ordinary manifold or like R x C (n points). TillS
'I' ( )
2. M is a closed V, n -manifold.
theory can be made into 'coefficients Z ' by adding orientations and an
i
n
j
3. There is a natural way of regarding W x M as a V-manifold,
untwisted neighbourhood for the singularity (see Chapter ill, Example; .. ,
j.
,.,oreach U-mamfold W (e. g. by relabelhng or forgettmg some structure).
1 1(2))
The twisted theory is interesting in connection with represen-
.•. Then the theory VIU x M is defined by considering polyhedra P
ting Z -homology (see §S).
I....
n .f1tha two stage stratIfIcatIOn P::) S(P) and extra structure such that:
3. £0
=
{p'l,
£1
=
{xix ~
SO
J,
£n is all closed £n-l-
i
1.
P - S(P) is a (V, q)-manifold.
manifolds.
1
2. S(P) is a (U, q-n-l)-manifold.
This theory is 'ordinary' homology with coefficients Zz' To
i
3. There is a regular neighbourhood N of S(P) in P and a
obtain coefficients Z one needs to orient the top stratum. We can think
II
isomorphism h: N ...•S(P) x C(M), which carries S(P) by the identity
of this theory as obtained from bordism by killing all manifolds except
)0
S(P) x (cone pt. ).
j
,~'
I
1
,
I
1f
I
I
Basic links
Suppose M and N are closed £-maJiitolds then M x N is in
general not an £-manifold. However it is one if we have:
£
*
£
c
£ (i. e. the join of two links is again a link).
n q n+q
Then, with axiom 4, we have cup and cap products. More
generally if £,
'JIT
and ~ are three theories and £
*
'JIT
c ~
then we
have cup and cap products from £ and
:JlL
theory to ~ theory. For
example, if S is ordinary bordism theory (i. e. Sn
=
{(n-l)-sphere})
then £
*
S
c
£ by axiom 2, so that, as remarked above, cup and cap
products with bordism or cobordism classes are always defined.
Products
the fundami:mtalclass of ;, manifold (amalgamation) is :ways defined).

P is then called a dosed (V/U x M, q)-manifold. There is an
obvious notion of V/U x M-manifold with boundary and hence we have
geometric homology and cohomology theories (V
/U
x M)* and (V
/U
x
M
4. h is an isomorphism of (V, q)-manil'ulds off S(P) (where
S(P) x (C(M) _ (cone pt.)) is regarded as a V-manifold by part 3 of the
data).
Notation. If U ==V then we collapse the notation to V
/M.
j
j
1
I
bordism WI' Then M x S(W
1)
is bordant to W by the V-bordism
1
WI - (nbhd. of S(W
1)).
1
aL
==O. A V-manifold has no second stratum.
j ~~
1m
L.
Let
!!!V,
S(W))be a V
/U
x M-manifold such that S(W)
bounds the V-manifold B. Form the product B x C(M) and attach it
to W x {I} in W x I by the identity on S(W) x C(M). This constructs
aV/UxM-bordismof W==Wx
{oj
to a V-manifold.
Proof. Consider the sequence with X ==pt. A ==0.
i
From now on we will, for notational simplicity, deal only with the
jhOmOIOgytheories. Exactly similar constructions will hold for the co-
homology theories.
Let n~O(,; Z
[t])
be the theory 'smooth bordism with Z[
t]
coefficients' defined by considering p-manifolds, where
p
is a fixed
reSOIUti~n O.f
~[t],
with a reduction to SO of the stable normal bundle
.iofeach llltrlllslC stratum (see the last two sections). By the universal
I
jcoefficient sequence, this theory is isomorphic to n~O(,)
@
Z[
t]
the
i
l
l'
t· f
n
SO
t
d .
j
oca Isa lOn
0
.'* a od pnmes.
i
From results of Wall [14] we know that all the torsion in n~O
lis 2-torsion and hence that n~O(pt. ; Z[
t])
is a free polynomial algebra
I 4
Iongenerators [M
1],
[M
2]' ... ,
moreover we can take index(M
1)
==1
,and index(M.) ==0, i::; 1: take M ==CP and to obtain index (M.)==O
; 1 1 2. 1
jsubtract an appropriate number of copies of (CP
)J.
Now define theories
I • 2
t,
i ==1, 2, ... as follows:
I
Let W be a V- manif old bordant to
P
by V
/U
x M-
Let W be a V-manifold then W x M bounds the V
/U
x M-
LX
==
O.
manifold W x C(M).
Ker
L
C
1m X.
Proof of 4.1. (Compare the proof of the universal coefficient
formula.) The spaces X, A, P,
Q
play no role in the proof, so we
ignore them.
X
L
a
.•.•fB
V .•.•
V+ ...•(V
/U
x
M.) + ...•fB
VI'"
i q q n
1
q n i q-
which the reader can check is a generalisation of the Vniversal
Co-
efficient Sequence. C. f. Remark 6. 1.
Remark 4.2. There is also a notion of killing a whole family
jM.) of elements simultaneously. In this definition S(P) ==u (S(P).)
1
i
1
and the neighbourhood of S(P). satisfies the conditions of the definition
1
but with M. replacing M. This is then a generalisation of the killing
1
used in Chapter lIT to define coefficients. 4. I becomes sequences like:
Proposition 4. 1. There are long exact sequences
X
a
==O. If (W, S(W» is a V
/U
x M-manifold then S(W) x M bounds
L
a'
W - (nbhd. of S(W)).
X
.•.•Vq(X,
A) ...• Vq+n(X, A) ...• (V
/U
x M)q+n(X, A) ...•Vq_l(X,
A).
Ker
X
elm
a.
If W is a V-manifold such that W x M bounds the
X
+
L
+
a
q-l .V-manifold W', then we can glue W x C(M) to W' along W x M to
...•Vq(P,
Q) .•.•
Vqn(p,
Q) .•.•
(V
/D
x M)q n(p,
Q) .•.•
V
(P
Q)
, . form a V
/U
x M-manifold with second stratum W.
in which the homomorphisms are defined as follows.
X
is mUltiPlicationl Corollar 4 3 S V" th d
___ ._... ~._~_.~ - ------- --1
Y •• uppose IS a rlllg eory an [M] is not a
by M followed by the identification of part 3 of the data..
L
is the iden~zero_divisor in V*(pt.) (i. e. ~ultiplication by [M] is injective) then
on Eepresentatives.
a
restr.!.c.ts to the see<.J~~~tratum~ e'
l---.- -----
a(P:J S(P), f) ==(S(P), fIS(p) etc.
1·1
(V/M)*(pt.) ~ ideal ge~::~~~~ by [M]
j
I
88 89

Theorem 4. 5. There is a natural equivalence of theories
Thus J is the geometric theory obtained from smooth bordism by intro-
ducing coefficients in Z[~] and then killing all the free generators
except CP. Note that by repeated use of 4.3 we have:
2
and then it follows from 4.4 that lj/(pt.) is an isomorphism since the
class of CP is the generator of both groups. lj/ is defined on genera-
2
k k
tors by the formula lj/«M, f), qt )
=
(qM x (CP ) , f
0
11 ),
where
2 1
q
€
Z[t] and qM means M labelled by q. We have to check that lj/
is well-defined The only non-trivial part is that if [W]
€
n~O(pt.) then
1/I«M
x
V,
f
0
11 ),
1)
=
1/I«M,
f), s(W)). I. e. that
1
(M x (W- index(W)(CP )n/\ f
0
11 )
2 1
J2
=
J1 /M , and inductively Ji
=
Ji-l /M..
2 1
J1
=
nSo( .
Zll
* "
[2
Finally let
Proposition 4.4. J*(pt.) ~ Z[~ ][t] where t has dimension 4,
and is represented geometrically by CP labelled by l.
---,------------ 2 -----
Nowlet K denote the theory ko*( ,)
®
Z[~], i. e. the localiza-
tion of real connected K-theory at odd primes.
lj/ :
K'"
J.
is zero in J (here index(W)
=
0 if n
*
4k, for brevity of notation). This
follows from:
Proof. Sullivan
[10],
using a method similar to Conner and
Floyd [3], has constructed a natural transformation Proposition 4. 7.
L[W]
=
0 if index(W)
=
O.
L
is as in diagram
4.6.
s :
n~O( ,) ® Z[~] .•• K*( ,)
such that s(pt.) maps [Mn] to 0 if n
*
4k and to index(M)t\ where
t is the generator of K*(pt. ) ~ Z[~][t], if n
=
4k. He also proves that
s induces an isomorphism
n~O( ,)
®
nSO Zr~][t] ~ K*( ,)
*
where n~O
=
n~O(pt.) acts on Z[t ][t]
by
®1
s
n~O
-+
n~O
®
Zr~]
-+
K*(pt.) ~ Z[t][t].
Proof. [W]
=
L;
a..W. in nS*O(,; Z[~]) where a..
€
Z[~] and
1 1 1
W. are monomials in the generators CP
=
M , M , .... However,
1 2 1 2
using the product formula for the index, we can read off index(W) as
a. where a. is the coefficient of (M )n/4, since all the other M.
1 1 1 1
have index
O.
It follows that a.
=
0, and we can bord W to p' in the
1
theory J by using bordisms like C(M ) x W., where W.
=
M W..
2 1 1 2 1
Remark 4. 8. There is a similar geometric description of con-
nected KU-theory given by a similar construction using complex bordism
and the Conner-Floyd map [3].
We will construct a natural transformation
1/1
in the commutative 5. MORE EXAMPLES
diagram Example
5.1.
Some theories which represent Z -homology.
---------~-- p ----
Let p be prime. Define a theory of singularities by the basic
links ~
€
£, ,
(p)
€
£, ,
(p) * (p)
€
£, , ••.
where (p) is a set with p
a
1 2
90 91

6. COEFFICIENTS IN A GEOMETRIC THEORY
In this section V denotes a general geometric theory, that is to
say, a theory with singularities, labellings and generalised orientations,
as in §§2, 3. We will explain how coefficients work for V-bordism. We
leave the reader to take care of V-cobordism (V-mock bundles) and to
formulate the appropriate Thorn isomorphism and duality theorems. This
Then an £-manifold is called an Euler space and can be thought of as a
polyhedron with 'even local Euler characteristic'. Note that manifolds
are Euler spaces and that Euler spaces form a ring theory. An Euler
space has Steifel homology classes, [12, 16], (defined using the
combinatorial definition of Whitney et al., see Halperin and Toledo [2]).
The triangulation of a complex algebraic variety is an example of an
Euler space (Sullivan proves this by a careful induction on dimension
using the fact that each stratum is even-dimensional).
Example 5.3. The Casson-Quinn theories.
Finally we mention some examples of geometrically defined (co)-
homology theories, which do not fit as described into the pattern of this
chapter. These are the theories whose coefficients are the surgery ob-
structions. For details of the definition see Quinn [5]. A 'manifold' in
the theory is a surgery problem with a reference space (corresponding
to fundamental group) and a boundary on which the problem is a homotopy
equivalence. Of particular interest is the theory (U
4
G/PL)*, which is
the Casson-Quinn theory corresponding to
1f
=
O.
Sullivan has shown
1
[10] that, at odd primes, this theory is isomorphic to K-theory, as in
§4. This raises the question of whether there is a convenient geometrical
description for (U
4
G/PL)* (or even for G/PL* itself) at all primes.
Also relevant here is the question of whether K-theory has a simpler
geometrical representation than that given in §4. Note always that, by
Chapter VII, all cohomology theories have some geometric representation.
even
l
and inductively
£ I-manifold with even Euler
n-
characteristic} .
£
=
{p'l,
£
=
{(q)/q
a
1
£
=
{p Ip
is a closed
n
points in it.
WI:-
call an £-manifold a po-polyhedron. This is a ring
theory (see §3), the ring closure of 'twisted Z -bordism' (Example
p
3. 1(2)). An orientation for an n-dimensional p-polyhedron is a generator
of H (P; Z ) ~ Z. The theory of oriented p<polyhedra represents
n
p p
Z -homology, in other words the natural maps T ( )
-+
H ( ; Z ) and
p
n n
p
Tq( )
-+
Hq( ; Z ) are onto. This follows from:
p
Proposition (see
r
6]). Let U be a connected ring theory with
U (pt.) ~ Z. Then U represents Z -homology if and only if
o
p -- ------ p ----------
Uq(L )
-+
Hq(L ; Z ) is onto, where L is the Lens space Sn/p of
n n
p -------
n ---------
arbit~arily high dimension n.
Nowthe generators of H*(L ; Z ) are
(l'
and
f3,
where
(l' E
H
2
n
p
is represented by the inclusion of L 2 in L , and
f3
E
H
1
is represen-
n- n
ted by L
U
Dn-1
-+
L , where the disc is glued on by the p-fold cover.
n-2
a
n
Note that the Bockstein of
f3
is
(l'.
Both
(l'
and
f3
are p-polyhedra and
the representation property follows.
We can modify T£ in various ways, still preserving its property
of representing Z -homology, for example:
p
1. Make stable restrictions on the normal bundles of the
strata. E. g. impose stable orthogonal or unitary structures. Note that
(l'
and
f3
have such structures.
2. The normal bundle of one stratum in the others can be
restricted. 1. e. we can restrict the freedom to 'twist'. The point is
that the group used in the construction of {3is Z not
L
as allowed
p p
for in the definition of a p-polyhedron. To make this restriction into a
ring theory restriction, we impose the restriction that the group for the
normal block bundle of a stratum of codimension
l'
+
q in a stratum of
codimension
l'
is the wreath product Z
rlJ
Z
n.
Z .,.
~I
Z
(1'
copies).
p pup P
Both these modifications are examples of restriction on the normal
block bundle system. For more information on the algebra behind p-
polyhedra see Bullett [1].
Example 5. 2. Euler spaces.
This theory was invented by Akin and Sullivan [16] and has inter-
esting properties. Define link classes by
92
93

Short resolutions
section is modelled on Chapter III, we follow the section headings of
Chapter III, explaining where the difficulties lie.
Remark 6.
1.
Coefficients
p
is an example of killing, as des-
cribed in §4. To make the notation fit with §4, let V
=
V
®
F (V-
I
a
manifolds labelled by elements of B ) and U
=
V
I8l
F. The trans-
a
1 1
formation U x L(r,
p) ...•
V is given by ignoring the label on the first
1 1
factor. Then (V, p)-theory is the theory obtained from V by killing
1
simultaneously the elements {L(r, p)lr
E
B
1.
1
1
Definition 6.2. Let
8
1
denote the circle with the non-standard
I
framing. V is a good theory if there is a bordism D of
8
1
to zero
lin V such that ~ is a bordism of M x
8
1
to zero for each
j
j[M]
E
V*(pt.).
Let
p
be a short resolution of an abelian group G. A V-manifold
j
with coefficients in
p
can be defined exactly as in III §I and the theory, Remarks 6. 3.
1.
For ring theories our definition of a good
i
enjoys all the analogous properties. In particular there is a universal ,;theory coincides with Hilton's, see
r
4;
1.
9]. For general theories Hilton's
coefficient theorem.
1
definition is equivalent to insisting that
T/
x
8
1
is cobordant to zero for
[each V-mock bundle
T/.
This is in fact sufficient to prove Theorem
I"
4(1),
;t
w:
W~'
:ota::~d:::~ thenweeaneompletetheeon,truetton
jOf L(b
2'
p) -
plug in the bordism D of
8
1
to zero wherever appropriate
1(inmost cases D
=
D
2
and the construction coincides with the old one).
,I
jFunctoriality
Longer resolutions The best result we have, for a general theory, is the following:
Exactly the same proof as III §4, using
Coefficients in a short resolution of an abelian
Theorem 6. 4.
Remarks 6. 5. The universal coefficient sequence is natural
!(andhence, usually, splits) in exactly the same cases.
[
The description of coefficients in resolutions of length :s 4 in
Chapter III is again an example of killing (made precise as in 6.
1
above).
I
igroup gives a notion of coefficients which is functorial
To make this work for a general theory we need to regard L(b.,
p)
x M ','(I) fo d th .
1,
r goo eones
as a (V, p)-manifold for each V-manifold M and each link L(b.,
pl.
NOW,l(2)on the categ f d' t f f b l' d dd t .
1
i
ory
0
lrec sums
0
ree a e Ian groups an
0
orSlOn
each stratum of L(b.,
p)
is a disc, so we can regard L(b.,
p)
x M as
i ---. '---------,-- ...- ------
1 1
igroups.
a stratified set with each stratum a V-manifold, and the only possible
problem comes from 'restrictions on the normal bundle'. In general
this problem is solved by endowing L(b.,
p)
with the universal restric-
1
tion, namely framings of each stratum which fit together in a standard
way (i.e. L(b.,
p)
is an object in the theory of framed manifolds with) Proof of 6.4.
1.
coefficients - ~table homotopy with coefficients). For the Q-stratum there IRemark 6. 3(2).
1
is no problem (framing is equivalent to orientation). For the I-stratum' 2. Step
1.
Coefficients are always functorial on the category
we have to frame each I-disc extending given framings near the ends. :offree abelian groups.
Orientation considerations imply that this is possible but there is non- This is seen as follows. Define 0' ( ; F), where F is a free
q
uniqueness - there are two possible choices for each I-disc. Finally ,abelian group, by allowing no singularities in the representatives (i. e.
for the 2-stratum, the non-uniqueness of framings of circles implies:labellings only) and codimension
1
singularities only in the bordisms
that the framing may not be possible. We now make some more precise :(this requires only the definition of O-links, which, as seen earlier,
statements.ialways holds). Now
{2' ( ;
F) is the same as
n ( )
®
F by exactly the
. q q
jargument of ill ~4, but with all levels of singularities reduced one step.
,
94
95

[16]
[15]
[13]
[14]
We leave the reader to check the details here; geometrically all that is
required is a construction (not unique) of I-links, which we gave above.
Step 2. Coefficients are functorial for odd torsion groups.
The idea here is to use the fact that 8
1
has order two to complete
the construction ofthe 2-links for a 3-canonical resolution. At the final
-1
stage we have S labelled by g
E
G and g has order t, t odd. We
have to plug in a bordism of g8
1
to zero. Take
!.;
1 copies of Sl x I
-1 -1
framed so S is at both ends and glue all the copies of S together.
Finally glue on one copy of 8
1
x I by one end. This constructs the
required bordism. The new singularity is labelled by the relation
g
+
g
+ ... +
g (t times).
Step 3. Coefficients are functorial on the category of direct sums.
Let G
=
F
@
G , where F is free and G is an odd torsion
_ 1 1
group. Define
SV'(;
G) by using the two definitions given above.
q - -
Precisely a generator of il" is the union of a generator of il ( ; G ).
q q
1
A 'bordism' is similarly a union of bordisms. Now let G'
=
F'
@
G'
1
and h: G- G' a homomorphism. Then h splits as
F
(II
F'
••
@~
\B
G } •.
G'
1 1
since there is no non-trivial homomorphism G
1-
F'.
This means that we can define h[M] by simply relabelling and
we never meet the problem of relabelling an element with singularities
of too high a codimension. Similarly bordisms can be relabelled. Thus
il" is functorial. That it is isomorphic to the correct group follows from
Steps 1 and 2. This completes the proof.
Products, Bockstein sequence and rings.
or
coefficients
The rest of Chapter III goes through with obvious changes in the
general case. The constructions are functorial under the same conditions
as Theorem 6. 4.
96
REFERENCES FOR CHAPTER IV
[1] S. Bullett. Ph. D. thesis, Warwick University (1973).
[2] S. Halperin and D. Toledo. Stiefel Whitney homology classes.
Ann. of Math. 96 (3), (1972), 511-25.
[3] P. E. Conner and L. E. Floyd. The relation between cobordism
and K-theory. Springer-Verlag Lecture Notes No. 28.
[4] P. J. Hilton and A. Deleanu. On the splitting of universal co-
efficient sequences. Aarhus Univ., Algebraic Topology, Vol. I,
(1970), 180- 201.
[5] F. S. Quinn. A geometric formulation of surgery. Ph. D. Thesis,
Princeton (1969).
[6] C. P. Rourke. Representing homology classes. Bull. Lon. Math.
Soc., 5 (1973), 257- 60.
P1
C. P. Rourke and B. J. Sanderson. Block bundles: Ill. Ann. of
Math., 87, (1968), 431-83.
[8] C. P. Rourke and B.
J.
Sanderson. Introduction to pi topology.
Springer- Verlag, Berlin (1972).
[91
D. A. Stone. Stratified polyhedra. Springer-Verlag lecture
notes No. 252.
[10] D. Sullivan. Geometric topology lecture notes. M. I. T. (1970).
[11] D. Sullivan. Geometric topology seminar notes. Princeton (1967).
[12] D. Sullivan. Combinatorial invariants of analytic spaces. (To
appear. )
R. E. Strong. Notes on cobordism theory. Princeton U. P. (1968).
C. T. C. Wall. Determination of the cobordism ring. Ann. of
Math., 72 (1960), 292-311.
N. A. Baas. On bordism theory of manifolds with singularity.
Math. Scand., 33 (1973), 279-302.
E. Akin. Stiefel Whitney homology classes and bordism. Trans.
Amer. Math. Soc., 205 (1975), 341-59.
97

V"Equivariant theories and
operations
Proposition
1. 1.
Suppose G acts on X. Then there exists K
jsuch that
I
KI = X and there is induced a good action on K.
1
.;
!
1
It follows from Proposition
1. 1
and the subdivision construction
!thatthere is a G-homotopy functor T~ on G-polyhedra defined by con-
!sidering only good actions. In fact we have:
There is a natural isomorphism
Proposition
1.
2.
Proof. It follows easily from definitions that X/G has a
pl
jstructure so that the quotient map q : X - X/G is pl. Choose
11LI=X/G, so that for each g EG thesubpolyhedron q{xlgx=x}
a
EK if and only if q(a) EL then
A mock bundle
~IK
is a G-mock bundle if there is an action of
jG
on
E(~)
which induces an action on K. Let T~(K) denote the group
jofG-cobordism classes of G-mock bundles over K. If the action on
IE
W
is good (i. e. if g{3a=
p
implies g
I
(3
=
id. ) then the subdivision
, a a
iprocess can be carried through equivariantly simply by subdividing all
lblocks in an orbit isomorphically.
In §l of this chapter we describe a further extension of mock jis a subcomplex of L. Then define
bundles, to the equivariant case - the theory dual to equivariant bordism
-IK
is the desired complex.
and in §2 give a general construction which includes power operations
and characteristic classes. The remainder of the paper is concerned
with the case of Z - operations on
pl
cobordism. In §3we expound the
2
'expanded squares' ('expanded' rather than the familiar 'reduced' because
of our indexing convention for cohomology) and in §4we give the relation
with tom Dieck's operations [5]. §5describes the characteristic classes
associated to Z -block bundles and in §6we give a result inspired by
2
Quillen [3] which relates the total square of a mock bundle with the
transfer of the euler class of its twisted normal bundle. This leads, in
some cases, to the familiar connection between characteristic classes
and squares. Finally in §7we give an alternative definition of squares,
based on transversatility. This is like the 'internal' definition of the cup
product (see II end of §4).
1. EQUIVARIANT MOCK BUNDLES
Proof. Let [UL] ETq(X/G), with L as in the proof of
1. 1.
Let G be a finite group and X a polyhedron. By a G-action on !Then q*~ has a natural G-action and we can define
e[
~IL]
=
[q*
n
X we mean a
(pl)
map G x X - X satisfying
!g
is easily proved to be an isomorphism.
(i) for all g , g EG and x EX, g (g x) = (g g )x. From
1.
2 it is easily seen that the work of II carries over to the
1 2 1 2 1 2
(ii) if e EGis the identity then ex = x for all x EX. :equivariant case when the action is free, for example there are cup and
If X, Yare G polyhedra then a map f: X - Y is a G-map if icapproducts and Thorn and Poincare duality isomorphisms.
f commutes with the G-action. We then have the concept of equivariant' If U is a geometric theory then there is also a notion of equi-
bordism of X by equivariantly mapping G-manifolds into X and we show!variant U-bordism and equivariant U-mock bundles. We omit details.
below how to define equivariant cobordism via G-mock bundles. We shall see in the next section that the case of a G-mock bundle
I
Suppose now G acts on X
=
I
K
I.
Then we say G acts on K
i~1K
with the action in K not good is extremely interesting as the power
provided for each
a
E K and g EG,
ga
EK. The action is good if in ;operations spring from consider such cases.
addition whenever ga
=
a
we have g/a = id.
98
99

i
,~
'1
suppose given a non-trivial homomorphism
2. THE GENERAL CONSTRUCTIONAND THE POWER OPERATIONS
Let W be a free G-polyhedron and JaG-complex with fixed
point polyhedron F
c
I
J
I
=
X.
Let U be a geometric theory. We have the following commuta-
tive diagram of homomorphisms:
(2. 1)
J.l :
G -
2,.
Define
r
i
j8 :
Zq(K) -
zcg(~),
where Z denotes isomorphism classes of U-mock
: bundles,
r
---------
by
sW
= ~
x ... x ~ with G action given by permuting factors via
J.l.
1S
commutes with ~ubdivision of K and can be seen to define external
I _
;power operations
PI
(J.l,
W) : Vq(X) _Uqr (Xr
X
GW)
P
(J.l,
W) : Uq(X) _Vqr(X x W/G)
o
lby
P.(J.l,
W)
=
<1>.
0
s. When 11, is defined (see above) then we have
J
1 1 .
q qr+n
linternal operations
p!
(J.l,
W)
=
<I>!
0
s :
V
(X) - U (X).
THE EXPANDED SQUARES
Now restrict attention to ordinary pi cobordism (denoted
iT*( , ) as usual) and, in the construction of the end of the last section,
:let G
=
Z
= ~
and let W
=
Sn, the pi n-sphere with antipodal action.
122
:We
obtain the external and internal expanded squares:
i
J'
1
I
There is a relative version got by replacing J, Q, and F by
(J, J ), (Q, Q ), and (F, F ) respectively.
000
The construction of
<P , <l>
can be made for more conventional
o
1
types of bundles, for example vector bundles, spherical fibrations.
13.
Here
<l>
1 (~)
=
e-
1(~
x lW) and G acts on X x W by the diagonal
I,
action. r* is restriction and 11! is composition with the trivial mock
bundle with fibre W/G, when this is defined, that is, when M x W/G
is a V-manifold for each V-manifold M, (e. g. if W/G is aU-manifold
and
V
is a ring theory).
The whole diagram is natural for subdivisions of G-bundles over
for G inclusions J
C
J and W
C
W. Further
<l>,
depends only on
o
0 •
the free G-cobordism class of W.
Example
2.1.
Let u;X be a G-vector bundle with G
=
Z and
2
let W be the sphere Sn with antipodal action. Then uIF
=
U
o
@
u
1
where G acts trivially on the fibres of u
1
and antipodally in the fibres
of u. It easily follows that
<l>
(u)
=
1I*(u)
®
1I*(l )
+ 11*(U) where
l
o
0 10
2n
11
n
is the canonical line bundle on P and 11 and 11 are the obvious
n
1 2
projections.
'for n
=
0,
1,
2, ...
i
i
Remark 3.
1.
The name
I
expanded square
I
gains more credence
i ••
2r+n
ifrom the observation that we can choose representatives ~ for
iSq~+rr
{l
so that
I •
Power operations
Now let
L;
denote the symmetric group on r symbols and
r
;andfrom definitions we have [~2r]
=
[{l
u
[{l.
A similar remark
100
101

k
and Sq! (~
U 1/)
is illuminated
There is a commutative diagram
Corollary 3. 6.
,weget sq~([~]
U [1/))
=
sq~[~]
U
.Sq~[7]]..
The relation between Sq~~ u Sq~
TJ
Jbythe following.
I
I
AbUSingj
!
n n n+r .
Sq , Sq and Sq, _ar_e_h_om_o.._m._o_r_p_hl_s_m_s_.
1 0 .
It is sufficient to show Sqn is a homomorphism.
1
Proof.
Proposition 3. 3.
Lemma 3.2. (a) Sq~l{]
= [{]
u [{].
(b) i*Sqn[~]
=
r~]
x
[n
where i :
X x X x
pt.
-Xxxx
Snis;
1 -- Z2
the inclusion induced by the inclusion of a point in Sn.
(c) Let ulK be a block bundle and let tu be its (canonical)
Thorn class then Sqn(tu) is the (canonical) Thom class of
------- 1 -------------
E(u) x E(u)
Xz
Sn
-X
x
X
Xz
Sn.
2 2
(d) sq~+m[Mm]
=
[M x M
X
z
Sn] wher~ M is a closed mani- )
• 2
fold here regarded as an m-mock bundle over a point.
the notation we have
applies to
Sq~rn
i
=
0, 1.
I
The following lemma is easily proved.
m
which the square is a pull back.
!
i
The last factor is a composition (of mock bundles) Proof. This follows from 3. 5 and the diagram
~ x 7]x Sn - K x K x Sn - (K x K x Sn)jZ
2'
but the O-mock bundle K x K x Sn - (K x K x Sn);Z gives the class
o
2
o
E
T «K x K x Sn);Z ). The result follows.
2
The following is immediate from definitions.
103
Proposition 3.4. Let i: «K x L) x (K x L) x Sn);Z
- 2
- (K x K x Sn)jZ x (L x L x Sn)
IZ
2 2
102
be defined by' i [x , x , y , y , z]
=
([x
o' Yo'
z], [x , y , z)). Then
-----, 0 1 0 1 1 1
sq~r~ x 7]]
=
i*(Sq~[~] x Sq~[7])) for any ~, 7].
~ which each square is a pull back.
1
Proposition 3.
7.
j*SqnSqm[~]
=
Sqmsqn[
n
where
j
0 0 0 0
r
Q x Pn x Pm - Q x Pm x Pn is given by j{x, y, z)
=
(x, y, z).
j
i
Proof. This follows from commutativity in
1
, AXA
Corollary 3. 5. sqn and Sqn are ring homomorphisms.
:1·:
(Qx P )
x
P
--+-
«Q
x
Q
x
Sn)jZ x (Q
x
Q x Sn)jZ x Sm
x
Sm)jZ
1 _ 0 __ ~~ ~ __ n m 2 2 2
I
i
IIS~
j
IIS!k
Proof. It is sufficient to show that Sqn is a ring homomorphism,
1
I
AXA
but from the formula in 3.4 and the commutative diagram . (Q
x
P )
x
P ~ «Q
x
Q
x
Sm)jZ x (Q
x
Q x Sm)jZ x Sn
x
Sn)jZ
, m n 2 2 2
I
I
I

where k shuffles the spheres and each
A
is a suitable diagonal map. Proposition 4. 2.
(i)
i i
IJSq!W
=
Sq
IJW,
where on the ri~ht we
4. RELATIONS WITH TOM DIECK'S OPERATIONS have the usual Steenrod operation.
(ii) IJSq~W
=
IJRiW.
IJ[P .]
=
0 unless n
=
i.
n-l
In [5] tom Dieck defines operations in the smooth case analogous Proof. (i) follows from the axiomatic description of
to our Sq and Sq. That the definitions agree in the smooth case
o
1
from (ii), which comes from the fact that
follows from 3.2. c. We now look at the relationship between our internal
operation Sq~ and tom Dieck's internal operation. By virtue of the Tholl1
isomorphism, see e. g. [1], one readily proves
Sqi and
5. CHARACTERISTIC CLASSES
T*(X x P )
S'"
T*(X) @T*(P )
n n
and
where x is the euler class of the canonical line bundle
l;P
and by
n n
direct construction we have p : P 1 - P , the usual inclusion, and the
. x n- n
projection of Xl is the inclusion P ....• P. Thus
n-l n
Sqn(~q)
= ~
Rq+i~q @xi
o \
i=O
+. \
and the Rq
1
are tom Dieck's internal operations, with a change of sign
in the indexing.
In this section we present a special case of the construction of §1.
Let u/K be a block bundle with involution f: E(u) - E(u) satisfying
fp-la
=
p-la and IK/
=
{x : f(x)
=
x
l.
Recall that the inclusion
X
=
I
KICE (u) is the projection of the Thorn class tu and by virtue of
the involution we have [tu]
E
T"Gs(E(U),E(u)),
G
=
Z2'
Nowwe may
apply the construction of §1to get
W
(u)
=
<I>
[tu]
E
T-s(X x P ),
-n
0
n
and
, n-s
W' (u)
=
<1>,
[tu]
E
T (X).
n-s .
charac-
are the Z
2
W (u) is the euler class e(u
®
l ),
and
-n -------
n--
I I ,
The classes W· (u),
w·
+1(u), ... , W· + (u), ...
-s -s -s n
teristic classes of (u, f).
Denote by u
@
l
the block bundle
n
E(u) x
z
Sn - X x P n' In
2
the case that u is a vector bundle with antipodal action then u
®
l
n
coincides with the usual tensor product with the canonical line bundle.
From definitions we have:
Sq~+qw
= ~
[P .]R-q-i~.
. i=O
n-l
Consider
Proposition 4. ~•
- p,
T*(X)
®
T*(P ) ...•T*(X x P )
-+
T*(X).
n n
Proof.
Then
PI
xi has projection X x P ....• X and therefore
PI
xi
=
[P .]. 1.
. n-1 .' n-1
Further if UX has projection p
t :
E(~) ...•X then ~
®
Xl
has projec-
., Proposition 5. 1.
tion E(~) x P . - X x P and composing with X x P - X we see that
I
s
i n-l n n W· (u)
=
e(u).
PI ~ ®
x
=
[P
.H.
The result follows. -s
. n-1 To get some geometric insight into the meaning of the
Nowlet IJ: T*(-) - H*(-; Z ) be the Steenrod map (the identity
2
i consider the diagram
on representatives, see IV 3. 1(3)). Let Sq be the usual Steenrod squarE
(with a change of sign),
104
105

i
(i*W
u e(w )) = j*j,
W
for ~
E
Tq(x)
I! 0 1 .
Lemma 6. 2. Suppose given a block bundle w
IX
and isomor-
w @w !Y ~ w
Iy, y
C X. Then
o
1
i
Y
1
)0
E(w)'
t
i
1
~ jl
X
)0
E(w)'
Proof.
Consider the diagram of inclusions
Let
uIX
be a
pi
bundle with fibre some euclidean space, and
suppose A: X x P ..•.(X x X x Sn)/Z is given by A(x, [y
])=[
(x,x, y)].
n
2
Then there is a bundle inclusion u x P - A*(u x u x Sn/Z). Define
+
n
2
u
IX
x P to be the complementary block bundle. The existence and
n
+
uniqueness of u follows from 5.
1
of [4].
Now suppose given a mock bundle ~q;K with
I
K
I
=
X and
E~C Eu where u is a
pi
bundle with fibre Rq+mand for each
-1
0
-1
0
aE
K, P~
(a)
=
pu
(a)
n E~ and further suppose given a
pi
normal
-1 -1
G
-1
0
-1
bundle v
IE ~
for the inclusion so that Pv p~
(a)
=
Pu
(a)
n Pv (E~). In
this situation we say ~ is in u with normal bundle v.
Theorem
6.1.
Suppose ~q
IX
is in uq+mwith normal bundle
m
+
n
+
v . Then e(u )SqoW =p!e(v ), where p: E~ x Pn ..•.X x Pn is
given by p(x, y)
=
(p~(x), y).
is the diagram of inclusion, and
E(u)'
denotes the pair
(E(u),
E(u)).
We need the following generalisation of the clean intersection
formula, 3. 3 of [3].
where
C
C
x
P(v) denotes the mock bundle E(v)/Z - K. We have then:
2
In the above situation, W' (us) is the class of
--------~ l' ------
E(W
(u))-
-n
t
Xl ::
r+s .
-11
I
f : K x S - E(u) WIth f K
=
P(v),
Proposition 5. 2. Let gS;K be the trivial block bundle then
W' (gS) is the class of the projection K x P - K.
l' -------~-~-- l'
Now suppose vr+
1
;K and uS;K are block bundles with involution phism
dthtth
°
0 0
t" h
O
SI1> r+l~ r+s+l h --
an a ere IS an equlVarlan Isomorp Ism u '" v
=
g were
g is the trivial block bundle with standard involution and the involution on
s r+
1 " .
s r+
1
I
u @
V
IS mduced from u @v ~ p*v E(u). Then we have
u
K x gr+s C E(gr+s+
1)
and composing isomorphisms and the projection
into E(u) we get an equivariant transverse map
where
where p is the projection of
<l>
(tu) after subdivision so that
1
X x P C (E(u) x Sn)/Z appears as a subcomplex. From the diagram
n
2
we see that we may regard E(W (u)), after dividing out the Z action,
-n
2
as f-l(X) where f is an equivariant approximation to X x Sn-XCE(u)
which is transverse to X. Such a map f may be produced by using the
pi
transversatility theorem and an induction over the cells of K x Sn,
where Sn has a suitable equivariant cell structure. This gives an alter-
native definition of the characteristic classes.
From the alternative definition we have
Proposition 5. 3.
P(v) •.•.
IKI.
6. SQUARESAND EULER CLASSES
The purpose of this section is to prove Theorem 6.
1
below and
derive consequences. The result was inspired by Theorem 3.
12
of
Quillen [3].
106
107

Since i =
l l
we have j*i,
W
=
l *l
*j,
W.
Now apply (II;2.4) to the
1 0 1 •
o.
square,
p : E x P -X x P and p': (E x E x Sn)jZ -(XXXXSn)!Z
~ n n ~ ~
2 2
be the projections. Now apply 6.2 to the top square to get
Now from definitions i!p! = C*i! and so we have
~;i;
1
=
i,
PI
e(v
+) ....
(1)
...
Now apply 6.2 to the element p~(l) and the bottom square to get
and get
l *l
i i*W. Again apply (II;2.4) to the square
o
I!
0
1
l
E(w )'
1
•. E(w @w)'
r·
o
1
t
lo
10
i
Y
1
•
E(w)'
1
and get i i*i i*W, which is i (i*W u e(w )) by (II;2. 6).
11
0 0
1
I!
0
Proof of 6.l. Consider the diagram
Y
• X
since from definitions ~*p; (1)= SqnW• Since i;P; = c'*i; and
x-. a ...
+
~ c=c'~u we have from (1)and (2)that iIP,e(v+)=j,(sqnWue(u))
v •. a
and hence the result since
i
I is the Thorn isomorphism.
Suppose now that u, v are vector bundles. Then u, v have
underlying
pl
structure (see [2]),and it is easy to see that u+ = u ~
l
+
n
and v = v ~
l
(see for example [1;p. 138]). From 5. 1 we now have
n
the following corollary.
Corollary 6.3. If ~ is in u with normal bundle v and u, v
admit vector bundle structures then
Apply 6.2 after replacing E ~
C
Eu
-+
X by
Proof.
id
-+
E .
u
XCE
u
Corollary 6.4. For
pl
bundle u/x with (canonical) Thorn
class t , Sqn(t ) = i,e(u+), where i: X x P
-+
X x E(u) is the
--u au. -- n --
inclusion.
7. THE TRANSVERSALITY DEFINITION OF THE EXPANDED SQUARE
The previous section raises several interesting questions, e.g.
the relationbetween u+ and u ~
l
in case u is a
pl
bundle with
n
Z
-action. This section clarifiesthe situation, see e.g. Proposition 7. 2
2
E (E
x
E
x
Sn),
jZ
v
tV 2
C'
E (E xExSn),
jZ
u u
2
U
j'
n
i'
__________ ..••._ (X xXxSn)jZ 2
E~
xp
~~
n
n i
~
E(E
x P )'
v
v n
t
c
~
E(E
x.P)'
u
u 'n
uj'
~
xxp
x
n
where c, c' are collapsing maps. Let
108 109

If u is a block bundle which admits a Z - action
- -------------- 2 ---
Wn(u) ~s independent of the choice of Z2-action, and
!!
u reduces to a
pl
_b_un_d_l_e_t_h_en_
(i)
(ii)
+
e(u )
=
e(u
®
l )
=
W (u).
n -n
Proposition 7.2. Suppose that u is a block bundle with Z -action.
n
2---
Then Sqotu
=
i, W (u), where i: X x P - E x P is the inclusion
,-n ~
nun •
Proof. Consider ~ in
eN.
Then the quotient map
f: (E ~ x (X x RN~x x RN) x E~) x S~- «X x RN) x (XXRN)X Sn);Z2
is self transverse with self intersection2:\E ~ x E ~ x Sn);Z2' which gives
Sq~~ after restriction to X x P. But the restriction of the domain of
1.
n
f to X x P is E(v ) and its self intersection is P
(1
x t) The
n c c
P
<,.
n
result follows from the commutativity of the subdivision with the opera-
tion of making self transverse.
We shall use the new definition to prove:
Proof. It follows from
7.1
that Sqntu can be obtained by making
n
0
X x S - E(u) x P self transverse. On the other hand e(u
®
l )
is
n n
obtained by making X x P x SO- X x P - E(u
®
l )
self transverse
n n n •
Both E(u) x Pnand E(u
®
l
n) are quotients of E(u) x Sn under a (free)
Z2-action, by definition. Inductively make X x Sn - E(u) x Sn equi-
variantly (with respect to the diagonal action on the right) transverse to
X x Sn. Suppose the result is f: X x Sn - E(u) x Sn. Then
-1
n
I' :
f (X x S );Z - X x P is the projection of (u
®
l )
= W (u). On
2
f
n n -n
the other hand X x Sn - E(u) x Sn - E(u) x P is self transverse and
n
restricting to the intersection gives f' again.
below. This is achieved by a further definition of sq~ as a special case
of a general construction which we now describe.
-
Let c : X - X be an r-fold covering map. We define a function
P : Tq(X) - Trq(X) as follows. Let w
t'X
be the vector bundle with
c c
fibre at x
E
X the vector space with basis set c
-1
(x). Now let
[~] E
Tq(X) then for some large N we may assume ~ is in the trivial
bundle
eN
with normal bundle v, say. Define ~ in
e®
w by
c c
E(~ )
=
{y
®
z : y
E
E(~), c(z) = Pt(Y)
l.
Observe that ~
=
c, c*~. Now
c
1.
<,
C
0
define a (vector) bundle v /E(~ ) by taking the fibre over Y
®
z to be
c c
the subspace of the fibre of
e
®
wc at p~(y) generated by vectors a
®
b,
with c(b) = Pt(Y) and b
*-
z. Further define f: E(V1.)-
E(e ®
w ) by
<,
c c
f(y
®
z, ~a.
®
bo) = Y
®
z
+
~ao
®
bo. Note that the image of f is a
1 1 1 1
closed subspace of
E(e ®
w). Now suppose ~ has base
K,
so
IKI
=X.
1. C
Over a vertex of K, E(v ) falls into r-distinct pieces and under f these
c
pieces intersect in the fibre of
€
®
w over the vertex. Shift f so
c
the r-pieces and all the intersections are pairwise transverse. This can
be done using (TI;4.
1).
The total intersection is the block of P (~ )
c c
over the vertex. This is the start of an inductive process which we call
'making f self-transverse'. The resulting self-intersection is the total
space of P (~). The self-intersection should not be confused with the
c
result of intersecting f with itself, which is got by taking two copies of
f and intersecting one with the other which would give the cup product
[f] u (f] where f is regarded as a mock bundle over
E(e ®
W ).
c Corollary 7. 3.
It is not hard to show that the operation is well defined and functori-then
al for bundle maps, and for the trivial r-fold cover P is just the --
c
r-fold product (see II end of
§4),
The description of P (~) considerably simplifies in case
c
p ~ : E~- X is an embedding. In this case one can inductively make
p : E(c, c*~) - X self transverse. As an example consider the double
cover °c : Sn _
P
and p t :
P 1_ P
the inclusion so that [~] ET-1(P ) Proof. (i) is immediate and (ii) follows from 6.4 and 7.2.
n
<,
n- n
1
n
is the generator. Then P (~)
=
0 since Sn- _
P -
P can be shifted Let Xmbe the class of the inclusion
PIC P
and let
c n-l n m- m
to have empty self intersection. H(m-l, n)
C
Pm-l x Pn
C
Pm x Pn be defined by {(x, y) : LXiYi=O },
see e. g.
[1]
p.
164.
Theorem 7.
1.
Let
n
then Sq ~
=
P (lpx ~).
-- 0
c
n
c : Sn x X - P x X be the double cover,
n --------
110
111

Corollary 7.4.
H(n, m)
C
P x P •
n m
Sqn(X ) is the class of the inclusion
o m ------------
VI·Sheaves
REFERENCES FOR CHAPTER V
In this chapter we extend the treatment of coefficients in Chapter
III to cover sheaves of abelian groups. We work always with pl co-
bordism but everything that we say can be extended to an arbitrary theory
under the conditions of IV 6.4. §§4 and 5 in fact extend unconditionally.
The general definition of sheaves of coefficients does not have all the best
properties one would hope for and we will explain where the difficulties
lie at the start of §4.
In §l we recall the basic properties of stacks and sheaves and in
§2we define the theory of mock bundles with coefficients in a stack. The
definition is functorial on the category of all stacks of abelian groups.
The main theorem asserts that, if the stack is 'nice', then there is a
spectral sequence expressing the relation between simplicial cohomology
and cobordism with coefficients in the stack. In §3 cobordism with co-
efficients in a sheaf is defined by means of a simplicial analogue of the
Cech procedure. In §4we discuss an extension of the methods used in
the previous sections and give an example of 'Poincare duality' between
bordism and cobordism with coefficients in the sheaf of local homology
of a Zn-manifold. Finally in §5we extend the methods further and give
examples which suggest the existence of a bordism version of the Zeeman
duality spectral sequence [1].
t
l.
So
m-l
H(m-l, n)
C
P x P by direct
m n
p. 164.
Proof. X is essentially the Thorn class
m
Sqn(X )
=
i,
e(l
1 i8ll ) but this is
om. m- n
construction, see Theorem 6.6 of [1],
[1] T. Brocker and T. tom Dieck. Kobordismen Theorie. Springer-
Verlag lecture notes no. 178.
[2] R. Lashof and M. Rothenberg. Microbundles and smoothing.
Topology, 3 (1965), 357-88.
[3] D. Quillen. Elementary proofs of some results of cobordism
theory using Steenrod operations. Advances in Math., 7 (1971),
29-56.
[4] C. P. Rourke and B.
J.
Sanderson. Block bundles ll: trans-
versality. Annals of Maths., 87 (1968), 255-77.
[5] T. tom Dieck. Steenrod-operationen in Kobordismen-Theorien.
Math. Z., 107 (1968), 380-401.
Definition 7.5. For any block, vector or pl bundle u define
W (u) by i, W (u)
=
Sqn(tu), where i: X x P - E x P .
-n ,-n
0
nun
In view of 7. 2 no confusion can arise from this definition.
1. STACKS
A stack of abelian groups over a cell complex K is a covariant
functor
T: ~ -
<tb, where <tb denotes the category of abelian groups
and ~ denotes the category with objects the cells of K and morphisms
the face inclusions. A homomorphism between stacks,
T
IK,
T'
IK,
is a
natural transformation of functors
1/>:
T -
T'.
The category of stacks
over K will be denoted by S
IK
or simply S. If
T
is a stack over K
112
113

and K'
<l
K, the subdivision of
T
over K' is the stack
T'
/K' such that
T'(a')
=
T(a), T'(a' <a')
=
T(a <a)· a'
(r'
a'
E K'
a a a
EK
1 2 1 2' , l' 2 "1' 2 '
a'
C
a u'
C
a u' Ca.
If
T
/K is a stack, Tx I will denote the stack
, 1 l' 2 2
over the cell complex K x I, such that:
(T
x I)(a x 1) =
(T
XI)(a) =
T(a)
for every aEK- (TxI)(a xI<a XI)=(TxI)(u
<a)=
, 1 2 1 2
T(
a
<
u ), a , a
E
K. If
T
/K is a stack, and
J
C
K, the restriction
1 2 1 2
T
I.T
is defined to be the stack over
J,
such that
(T
I
J)(
a)
=
T(
a),
a
E
J. (1IJ)(a
<a )
=
T(a <a ), a , a
E
J.
, 12 1212
Let X be a polyhedron and F a presheaf of abelian groups over
X. If K is a cell complex,
I
K
I
= X, F induces a stack, FK/K, by:
FK(a)
= F(st(a, K»,
FK(al
< (
2)
=
F(st(a1,
K)::>
st(a2,
K»,
a, aI' a2
EK.
We briefly recall the notion of simplicial cohomology with coef-
ficients in a stack and its relation with Cech cohomology. Let T/K be
a stack over the oriented simplicial complex K. A (-p)-cochain, fP,
with coefficients in
T,
is a map which assigns to each p-simplex
~ E
K, an element of
T(uP);
(-p)-cochains form an abelian group by
coordinate addition, C-P(K,
T).
There is a coboundary homomorphism:
o-P : C-P(K, T)- C-p-l(K,
T),
given by
oPfP(JJ+1) =
L:
+1[a: uP+1]T(a<
~+1)fP(a). {CP(K,
T),
oP} is a
chain comPle:=:'~ its cohomology is by definition HP(K,
T).
If F
IX
is a sheaf and FK the induced stack over K, let ~
denote the covering of X formed by the stars of the vertices of K. The
nerve of ~ is known to be equ~ to K and this fact induces a canonical
identification {CP(K, FK), oP} .:: {CP(~, F),
6}
of chain complexes;
•.. stands for ,eech'. Therefore, because the coverings by stars are a
cofinal system in the system of all coverings of X, we get
limHP(K; FK) = limHP('U, F) = HP(X; F).
-+K
-'U
2. COBORDISMWITH COEFFICIENTS IN A STACK
Throughout this section we shall be using the same terminology as
in Chapter III below 3. 5. Thus a G-cycle or G-manifold has singularities
in codimension 1 at most, while a G-bordism is allowed to have singular i-
ties up to codimension 2.
114
Let T be a stack of abelian groups over an oriented cell complex
Ie.
A
(T,
q)-cocycl~obordism] ~q/K consists of a projection
p ~ :
E -
I
K
I
such that
(a) for each
a
E
K, p~l(a) is the interior of a
(T(a),
q+dima)-
manifold [bordism],
uaY
(b) for each
a
E
K,
1f<T)
= ~
[a
i:
a]T(a
i
< a)p~
1
(a
i),
where
1
r(a.
<
a)p~l(a.)
is the image of the T(a.)-manifold [bordism] under the
1 ~ 1 1
relabelling morphism
T(a.
<
a); [a. : a]
is the incidence number and its
1 1
effect is a change of orientation iff
[a. : a]
=
-1.
1
The manifold [bordism]
~(a)
=
uaY -
0
uaY
is called the block
~ver
a;
(T,
q)-cocyc1es [cobordisms] ~,
T/
over K are isomorphic,
written ~
2" T/,
if there is a
(Pi)
homeomorphism h: E ~ - E
T/
such
chat h
I
~(a)
is an isomorphism of T(a)-manifolds [bordisms] between
((a)
and
1]
(a).
Suppose given ~q/K and L
C
K, then the restriction
\ I
L is defined in the obvious way and it is a
(T
I
L, q)-cocycle [cobordism]
over L. A
(T,
q)-cocycle [cobordism] over (K, L) (L
C
K) is a
(T,
q)-
cocycle [cobordism] over K, which is empty over L; - ~ is the cocycle
cobordism] obtained from ~ by reversing the orientation in each block.
If ~ and ~ are (1, q)-cocycles over (K, L), then ~ is cobordant to
o
1 0 ~--~
\ if there exists a (T x I, q)-cobordism
T/
/(K x I, L x I) such that
1 .
~I
K xli} =
(_1)1~.
(i = 0, 1). Cobordism is an equivalence relation
1
Indwe define Uq(K, L;
T)
to be the set of cobordism classes of
(T,
q)-
:ocycles over (K, L); Uq(K, L, T) is an abelian group under the opera-
:ionof disjoint union; we call it the q-th oriented
(Pi)
cobordism group
)f (K, L) with coefficients in
T.
If T'
<J
T
then there exists an 'amalga-
nation' homomorphism am: Uq(K',
T') -
Uq(K,
T)
like in the case of
)rdinary mock bundles (see Chapter II). In the proof of the subdivision
:heorem for cobordism without coefficients, Chapter II 2. 1, all the
;eometric constructions are carried out cellwise. Therefore the proof
~eadilyadapts to the present case and we deduce
Proposition 2. 1.
!f. T'
<J
T
then am: Uq(K',
T') -
Uq(K, T)
.san isomorphism of abelian groups.
Let
¢:
T •• T
be a homomorphism of stacks over K. There is
1 2
Ininduced homomorphism 1Jl*:U*(K; T ) - U*(K, T) defined blockwise
1 2
115

q+dima)-
- 1
~(a)
=_
[a. : a]p(a. <
a)p~
(a.)
(with
a<a
1 1 ., 1
he usual meaning of the notations); i
(c) P-l(L) =
¢.
fwo
(p,
q)-mock bundles ~ , ~/(K, L) are cobordant if there exists a
o
1 .
p,
q)-mockbundle 17/(KXI, LX
o
such that TJIK. =
(_1)1~.
i=
0,1.
1 1
fhe (q-th)-cobordism group of (K, L) with coefficients
p,
written
)q(-;
p),
is constructed from
(p,
q)-mock bundles in the usual fashion,
Thus the main difference between the theory of
(T,
q)-cocycles and
he theory of
(p,
q)-mock bundles is that the former allows the cobor-
lisms to have deeper singularities than the cocydes while the latter is
he natural extension to the case of local coefficients of an ordinary
nock-bundle theory.
vhere g is induced by (K, L)'" (K, ,0)and f is 'restriction to L',
Lemma 2. 4. There exists a coboundary homomorphism
jq :
nq(L;
p) .••
nq-
1
(K, L;
p)
and a long exact sequence
g
f
oq
, ...•• nq(K, L;
p) .••
nq(K;
p) .••
nq(L;
p) .••
!!.
T
E
S', there is a spectral sequence, E(T),
n;(a)
=
nq(point;
T(a»
nq(a
<a ) "" T(a
<
a
)* '
T 1 2 1 2
We aim to prove the following
E
2
= HP(K' nq)
=}
n*(K-
T)
p,q
'T "
Theorem 2. 3.
by relabelling; more precisely, if UK is a T1-cocycle [cobordism] and
K,
L) consists of a projection p~ : E~'" IKI such that
a
E
K, we relabel the block
~(a)
by means of
cp(a):
T
(a) .••
T
(a)
(a) for each
a
E
K, p~l(a) is the interior of a
(p(a),
1 2 .,
(like in III). If
~!(a)
is the resulting polyhedron, then the union nanifold,
H<J);
~'= __
~'(a)
is aT -cocycle [cobordism]. All compatibility conditions (b) for each
a
E
K,
aEK 2
are ensured by
!/J:
T ••• T
being a stack-homomorphism. Therefore
1 2
we have the following:
Proposition 2,2, There is a functor n*(K, -) : S'" G.b* which
assigns to each
T E
S the abelian group n*(K;
T)
and to each morphism
If>
E
S the (graded) abelian-group homomorphism
cp*,
A linked stack of resolutions over a cell complex K consists of
a covariant functor
p: ~ ~ e
(see III §3 for the definition of
e),
A
linked stack of resolutions
p
is said to be p-canonical if
p(a)
is a p-
canonical linked resolution for each
a
E
K (in the sense of III §3).
If
T
IK
is a stack of abelian groups and
p
IK
is a linked stack of
resolutions such that, for each
a
E
K,
p(a)
is a resolution of
T(a),
then
we say that
T
is represented by
p.
We denote S'
IK
the full sub-
category of :ilK consisting of all stacks which are representable by
p-canonical stacks for each p
=
1,
2, 3, The objects of S' will also be
called nice stacks.
If
TIK
is a stack of abelian groups, there is an induced graded
stack on K, written {nq
IK)
and given by
T
running
Proof. Definition of oq, It can be roughly described as 'pull
lack onto the boundary of a regular neighbourhood of L in K', Suppose
L
full in K. Let
J(t)
be the
t
-nhd, of L in K and
j
its frontier;
i:
J'"
L the pseudo-radial retraction, ir
=
?TIt
If ~ is a
(p,
q)-mock
IUndleover L, form the pull back
ir*(~),
which is a q-mock bundle over
r
and a (q - I)-mock bundle over (K, L), If
~*(a)
is the block of
1i-*W
Iver
a,
then
~*(a)
comes from the block over
n-(j(t)
n
a),
denoted
:(a);
therefore it has a structure of p(ir(a»-manifold and it is made
nto a p(a)-manifold by means of the stack homomorphism
(p(ir(a)< a),
Sothe
'esulting object
ir*(~)1K
is a
(p,
q-l)-mock bundle which is empty over
Moreover the spectral sequence is natural on S', ,
.- -----.~-------- J,
The assignment [~] .••
[1T*WIK]
gives a well defined morphism
In order to prove the theorem we need some definitions and lemmlllq q() q-l
qf : n L .••n (K, L), If L is not full, one first subdivides bary-
Let
p
be a linked stack over K. A
(p,
q)-mock bundle ~ over
----- :entrically once and then amalgamates.
116
117

s
F n*(K;
p)
=
Ker[n*(K;
p) •.•
n*(Kp_1;
p)].
Proof of Theorem 2. 3. For each i
=
1, 2, 3, let
p.
be an i-
1
anonicallinked stack representing a given
T
f
S'.
There is a commutative diagram of degree-zero homomorphisms:
1
which t.. is the 'relabelling' map on cocycles. It is easy to see that
1,]
. commutes with the coboundary operation and therefore there is an
.,)
lduced commutative diagram of spectral-sequence homomorphisms
Lemma 2. 5.
!!.
p
is a linked stack of resolutions there exists
a spectral sequence
E(p)
running
Proof. By Lemma 2.4, for each p
=
0, 1, ... we have the
exact 'p-sequence'
Exactness is proved geometrically by mock-bundle arguments, .asa block
Ha)
for each p-simplex
a
f
K and
~(a)
is a closed
p
which are all contained in Chapter II. The only remark to make is the
i(a),
-q)-manifold, because ~ is empty over K l' Therefore
~(a)
p-
following. etermines an element
[~(a)]
f
nq(a). We associate to [~] a p-cochain
p p
Suppose K
=
a
and ~ is a
(p,
q)-mock bundle defined on ~;] with coefficients n~ by setting
J
=
a -
a ,a
face of
a.
Then
p
gives morphisms going from the
1 1
resolutions attached to the simplexes of
J
to the resolutions attached ff~](a)
=
[~(a)]p
for each p-simplex
a
E
K.
to
a.
Therefore, if ~ is extended over
a
by the pull back construction, p
:is readily checked that the correspondence [~] ..• f[] defines the iso-
the resulting block
~(a)
has a natural structure of p(a)-manifold. ~.
q lOrphism h. Moreover the required convergence conditions hold and
We remark that the coboundary homomorphism /) can be defined
00
( ) ( )aerefore E is the bigraded module associated to the filtration of
also for the theory n* K, L;
T T
f
S/K exactly in the same way as in I*(K;
p)
defined by
the above proof. Therefore there is, for the theory n*(K, L;
T),
a long
sequence analogous to that of Lemma 2.4. However, although it is
immediately checked that the sequence has order two, there is no reason
to suppose that it is exact. The argument which is used to prove Lemma 'he lemma follows.
2. 4 fails because in n*(K, L;
T)
two cobordisms having the same ends
cannot be glued together to give a cocycle.
Given a linked stack of resolutions
p
/K there is an associated
graded stack nq
f
S defined like nqabove
(T E
S). We then have the
p
T
following:
g
f
... n-P-q+l(K , K .
p) •.•
n-p-q+1(K .
p) .••
n-p-q+1(K .
p)
/) P p-l' p' p-l'
..• n-P-q(K K .
p) .••
p' p-l' ...
where K is the p-skeleton of K. The theory of exact couples yields
p
a spectral sequence
E(p)
in which the chain complex {E
1;
d
1:
p,q p,q
q fixed} is isomorphic to {CP(K;nq), /)} defined in Section 1. More
p
precisely an isomorphism h: E
1•.•
CP(K; nq) is given as follows.
1 _
-p-q . p, q -p-q
p .
E - n (K , K l'
pl.
Let [~]
f
n (K , K l'
pl.
Then ~
p,q p p- p p-
t*
E(p )
1,2
lO
E(p)
t;,~ /;,:
E(P3)
In
the E
2
-term t:+'. is the homomorphism
1,
J
t:+' .
1, ]
HP(K;n~.) ..•
1
118
119

Corollary 2. 7. If T
E
S' and we take the theory n*(K) to be
H*(K, Z) (= simplicial cohomology with Z-coefficients), then n*(K, T)
~~incides with the usual definition of simplicial cohomology with co-
efficients in T (see Section
1).
morfhism.
nq . relabel nq whl'ch we know E(G)
-+
E(T) is an isomorphism on the E
2
-term because n*T=n*(point, G).
induced by the change of coefficients"" ---".-""
Pi
Pj Therefore the corollary follows from the 'mapping theorem
to
lit!
an isomorphism from the theory of coefficients in the constant case'
b1 '
etween spectra sequences.
Then, by the usual spectral-sequence argument, tt . il:lan iso-
, J
morfhism for all (i, j) :
1
::si
<.
j
::s3. Nowthere is a homomorphism
(]I :
r.*(K; T)
-+
n*(K;
P )
given by the relabelling map on the cocycles.
3
In order to prove the theorem we only need to show that
(]I
is an iso-
1.
QI is an epimorphism. There is
~l
commutative diagram Proof. H*(K; Z) is the mock-bundle theory whose blocks are
oriented pseudomanifolds. Since a G-pseudomanifold of dimension
greater than zero is bordant to
¢
for every group G, we see that the
E1-term of the spectral sequence E(T) reduces to the cochain complex
C*(K, T) considered in Section
1
and the spectral sequence collapses.
3.
COBORDISM WITH CQEFFICIENTS IN A (PRE)-SHEAF
We are now ready to give a notion of cobordism with coefficients
in a presheaf, using an analogue of the Cech procedure. Let X be a
polyhedron and FIX a presheaf of abelian groups. If K is a triangula-
Therefore, since t is epi, so
1,1
HP(K, n*(point, G))
'-'?
n*(K; G).
t : n*(K; G)
-+
n*(K; T)
tion of X, then, by the previous section, we have a graded group
{nq(K, F K) }, where F K is the induced stack on K. Suppose K'
<I
K.
We
define a homomorphism QlKK' : nq(K, FK)
-+
nq(K', FK,) as follows:
let ~q
IK
be an (FK, q)-mock b~ndle. Subdivide ~ over K' and get
. . ~"IK'
suchthat
~"(a')
isanFK(a)-manifoldfor
a'ca.
Theinclusion
Corollary 2.6. If T/K IS a constant stack (1.e.
T(a)
= G for t(' K')
c
t( K)' t . t· h h'
- ----- •. u •••
sa,
s
a,
gIves a res rlC lOn omomorp Ism
each
a
E
K), then n*(K; T) co~ides with
i~*(K;
G) _asdefined i
l1
JL?,.
Faa' : FK(a)
-+
FK,(a')
and we make
~"(a')
into an FK,(a')-manifold,
,
~'(a'),
by applying this homomorphism, i. e.
f(a')
= F
,(~"(a')).
When
a,a
111
the blocks of ~" have been relabelled by means of the restriction
homomorphisms, one takes care of the orientations in the blocks, so that
Iheincidence numbers are preserved. The functoriality of the presheaf
F
ensures that the final object is an (FK" q)-mock bundle
~'IK',
called
in F-subdivision of ~ over K'. Two F-subdivisions ~',
1;'
of ~ over
[' are cobordant by the same construction applied to K x I and K' x I
modulothe ends. Therefore we have a well defined homomorphism
Proof. Since n* (K; G) is a cohomolo~.,'ytheory, there is a
spectral sequence E(G) running
2. QI is a monomorphism. Let [~]
E
Ker
a.
It follows from
the definitions that
~IK
is also a
(p ,
q)-mO(~kbundle. Therefore it
2
determines a class
[~]p
which is mapped to zero by t?,
3'
Because
t is a monomorphisJ.. there exists a
(p
,q)-cobordism
11 : ~ - (6.
2, ., 2
But
1]
is also a
(T,
q)-cobordism. Therefore (}' is mono.
The theorem follows.
There is a natural transformation of cohomolob'Ytheories
where
e
is also a relabelling map.
is
(\I.
given by relabelling. The induced homomorphism of spectral sequences
120
121

The collection of gr oups and homomorphisms
I
nq(K, FK1, O!K K'}'
indexed by the directed set of all triangulations of X, is a dire~t system
and we define the q-th
(pl)
cobordism group of. X with coefficients in F
to be the gr aded gr oup:
is a natural transformation of cohomology theories, inducing an iso-
morphism of the corresponding graded sheaves
h . T*
-+
S*
F' F F
then h induces an isomorphism in local coefficients:
h*(X) : T*(X; F)
-+
S*(X; F) .
4. QUASI-LINKEDSTACKS
The proof is the same as that of 2. 3, using limit stacks.
A direct consequence of the 'mapping theorem between spectral
sequences' is the following comparison theorem
We now recall that a sheaf
FIX
is locally constant if there is an We have given a definition of
'pl
cobordism with coefficients in
open covering
(i
=
{U) of X, such that, if U
E
(i
and x
E
U then a sheaf' which works in the category of all sheaves, is functorial on this
F(U)
=
lim {F(V)} where V varies over the open neighbourhoods of x. category and extends the case of coefficients in an abelian group. Un-
-+
It then follows that, if K is a sufficiently small triangulation of X, i. e. fortunately, we can be sure that the definition enjoys good properties
the associated star-covering is a refinement of
ct,
the cohomology of K (e. g. spectral sequence) only when some special types of sheaves are
with coefficients in the stack FK coincides with the Cech-cohomology involved. Let us try to describe the source of this difficulty. A mock
H*(X; FK) and the cobordism of K with coefficients in FK coincides bundle and a stack, T, over a simplicial complex K, have a main feature
with n* (X; F) by 2.
1.
We call such an F
KIK
a limit stack for F
IX.
in common: they are both 'local' objects in the sense that they are both
We say that a locally constant sheaf
FIX
is nice if it has a limit stack functors defined on the category ~. A mock bundle takes its values in
which is nice in the sense of Section 2. the category of pl manifolds and inclusions in the boundary, while a
As in the case of stacks, to each sheaf F
IX
there is associated stack of abelian groups ranges in the category of abelian groups and
a graded sheaf
I
n~IX)
defined by fl~(U)
=
nq(point; F(U)), homomorphisms. Thus in order to have a notion of mock bundle with
n~(U :) V)
=
F(U :) V)*; u, V open sets of X. If
FIX
is locally con-
T
coefficients, which has good properties, what is n~eded is the following:
stant, then
n~IX
is also locally constant.
1.
a notion of r(a)-manifold
(a
E
K) for which the corresponding
We have the following analogue of Theorem 2.3. mock bundle theory is 'cobordism with T(a)-coefficients'.
2. a recipe for associating a
r(a
I-manifold to a
r(a
I-manifold
Theorem 3.
1.
On the ca!e~()ry of nice sheaves there is a natural in a way which is functorial on K.
2 1
spectral sequence running h - .
~--_ ..-~~"". -~---~~ T erefore we see that the difficultIes ansmg from our defmition
can be traced to the lack of a bordism theory of G-cycles which is func-
torial (with respect to G) on the cycles, rather than only on the bordism
classes. Now in many instances it happens that conditions
1
and 2 can
be fulfilled and since some of the cases look rather interesting, we will
make a detailed discussion of them.
122
Proposition 3. 2. Thus, in the remainder of the chapter, we abandon the point of view
Let
F*IX
be a nice sheaf. If h: T*(-)-+S*(-)of setting up a theory of local coefficients in generality. Instead we dis-
123

sheaf'.
With the above notations we have the following
The sheet for which the assignment (i) holds (i
=
1, 2, 3) will be referred
to as 'sheet (i)'.
Associated to £. there is the following quasi-linked stack, A, on
, id
the Z -manifold K: if
a
EK - Sl.{Athen
A(a)
is F(x
T) -.
F(xa);
if
3
aala aa )
a
ESK then
A(a)
is F(x x)
>-40>
F(x , x );
A(a'
<
a)
=
£.(a'
<
a.
, l' 2 1 2
We write nq(p,.£)
=
lim nq(K, A). Here .£ stands for 'local-homology
..•
cuss, along the lines of the above remarks, some generalisations of our and nq(a
<a ) :
[M].••
[S(a <a
)M] •
s
1 2 1 2
method and give examples to show how local coefficients may at times As an illustration of the general setting described above, we dis-
reveal relationships between local and global properties of a space. We cuss some examples associated with Poincare duality.
continue to work only with pl cobordism, but everything that we say Let (P SP) be a Z -manifold of dimension m (see III 1. 1(2)),
, n
remains true for an arbitrary geometric cohomology theory. triangulated by (K, SK)(SK
C
K) and consider the stack
£
=
£'(K) of
Let
p, p'
be linked resolutions and f :
p •••p'
a chain map. f local m-homology on K.
is said to be quasi-linked if the following conditions are satisfied: If n
=
2, then a presentati?n of £. is the following:
(a) For each b
PE
B
P, P
=
0, 1, 2, 3 either f(b
P)
=
0 or (a)
£.(a)
=
free abelian group, F(X~, on one element xa
one of ±f(b
P)
is in
B'P.
(b) if
a', a
EK _ SK or
a', a
ESK and
a'
<
a,
then
(b) For each link L(b
P,
p)
let fL(b
P,
p)
be obtained from
£.(a'
<
a)
is the isomorphism mapping
xa'
to
xa;
L(~,
p)
by the fOllowing process. Let V
C
L(b
P,
p)
be a stratum (c) if
a'
ESK,
a
EK _ SK,
a'
<
a,
then
£.(a'
<
a)
is one of the
labelled by bj: if f(bj)
=
0, remove V from L(~,
p);
if isomorphisms x .••±x , depending on which 'sheet'
a
belongs to. We
j' j j j
a' a
f(b) EB'l, relabel V by f(b); if -f(b) EB', reverse the orientation call 'positive' (resp 'negative'Hhe sheet corresponding to the
,+'
sign
.. p p ---
of V and then label -V by _f(b
J).
Then we requIre fL(b ,
p)
=
<5L(<5fb
,f:.
(resp. ,_, sign).
for one of <5
=
± . For a Z -manifold, £. can be presented as follows. If
a
EK- SK,
3
a a
To each
(p,
n)-manifold M there is associated a
(p',
n)-manifold then
£.(a)
=
F(Xa); if
a
ESK, then
£.(a)
=
F(X
1,
x/ The morphisms
f(M) constructed on strata in the same way as fL(b
1,
pl.
There is a are so defined.
category
Q,
whose objects are short linked resolutions and whose mor- (a) if
a', a
EK _ SK or
a', a
EK and
a' <a
then
£.(a' <a)
phisms are quasi-linked maps. A quasi-linked stack of resolutions is the canonical isomorphism;
over a ball complex K is a covariant functor S
=
SK : ~ .••
Q.
If K is (b) if
a'
ESK, and
a
EK _ SK;
a'
<
a,
then
£.(a'
<
a)
is one
oriented, an (S, q)-mock bundle ~q over K consists of a projection of the following isomorphisms, depending on which sheet
a
belongs to:
p~:E~"'IKI such that, for each aEK, p~l(a) istheinteriorofa~l {xa' .••x {xa, .••_x {x~''''O
(S(a),
q
+
dim a)-manifold
WJJ,
with
oW1J
= ~
[ai:a]S(ai
<
alp ~ (a/
(1)
1
a
(2)
1
a
(3)
I
a' a' a'
The cobordism theory of (S, q)-mock bundles is developed exactly x
2•••
xa x
2•••
0 x
2•••
-xa
in the same way as the theory of cobordism with coefficients in a linked
stack of resolutions
(see
§2). In particular we have
(a) an abelian group nq(K, L; S), the q-th cobordism group of
K, L (L
C
K) with coefficients in S, depending only on S,
I
K
I, I
L
I.
(b) a spectral sequence running
where oq is the stack of abelian groups over K defined by
s
124 125

Proposition
4. 1.
There is a duality isomorphism:
Proof. n
==
2. In this case SP is an orientation-type singularity.
Let [~]
E
nq(K, A). The total space
EW
is a manifold (see Chapter II).
Moreover E(~) is oriented, because it is oriented over both K - SK and
SK and the action of
£,
makes the orientations of the blocks over the
positive sheet compatible with those of the blocks over the negative sheet.
Therefore we regard E(~) as an oriented bordism class and define a
'glueing map'
1/1
like in Chapter II. The proof that
1/1
is an isomorphism
reduces essentially to the proof of Poincare duality given in II and we omit
it.
n
2::
3. Again, for the sake of simplicity, we only discuss the
case n
==
3. The general case is dealt with using the same arguments.
Let [~]c nq(K, A). The block of ~ over a simplex
a'
E
SK is a
pl
manifold, each component of which is labelled by either
xa'
or
xa'
and
I 2
at most two non-empty blocks merge into it, so that no singularities are
created in the glueing process (see Fig. 18). Moreover, as in the above
case, the signs in the stack-homomorphisms ensure that the orientations
of the blocks are compatible in passing from one sheet to another across
SK. Hence the total space E(~) is an oriented
pl
manifold and the
operation of 'glueing up and disregarding the labels' gives the required
homomorphism
1/1.
We nowprove that
1/1
is an epimorphism. Let f : Wm+'4K be
a simplicial map representing an oriented bordism class of P. Then
Cohen's generalized transversality theorem (see IV
1. 1(2)),
together with
the subdivision theorem, tells us that there exists a cell decomposition
(L, SL) of (P, SP) and
f""
f, such that
(a)
fir
1(L - Int L ) is the projection of a
pl
oriented q-mock
a
bundle, where L is a subcomplex of L triangulating a product neigh-
a
bourhood of SP in P, i. e.
I
L
I
=
SP x cone (3 pts). In order to avoid
a
technical details, we assume that L is the product cone-complex
. a
SL x cone (3 pts) and write L
1
for L n Sheet (i) (i
==
1,
2, 3).
a a
126
Fig.
18
(b) rl(L) is an oriented manifold with bOWldary rl(aL )
a a
and
T
Irl(L ) is the projection of a q-mock bundle over the cone com-
o
plex L
=
{a
x cone (3 pts) :
a
E
SL
l.
a .
Nowwe take boundary collars rl(aL
J)
x Ie rl(L )
(j
=
2, 3)
a a
and 'stretch' them along the cone-lines of L until
f
is replaced by a
m+q a
map f' : W ...•
1
L
I
such that, with obvious meaning of the notations:
(a) f' ""
T
mod L - Int L
. a .
(b) f,-I(SL)
=
rl(aL
J)
x
1
c r1(aL
J
x I) (j
=
2, 3). (See
a a
Fig.
19.)
Then, for
a'
E
SL and f,-l(a')
"*
li!,
we label the block f,-l(a')
a'
by xl if the two non-empty blocks merging into it lie over the sheets
(1)
and (2) respectively; otherwise we label f,-l(a') by
xa'.
For each
2
a
E
L - SL we give f,-l(a) the label x. In this way f' : Wm
+
q...•IL
I
a
gives rise to a q-mock bundle associated to AL, This shows that
lJ;
127

(*)
0
(SM)+
II(SM,
M)
=
0
(M)ISM,
a~(a)
=U
[a: T]S(T, aH(T)
T<
a
5. A FINAL EXTENSION AND EXAMPLES
where
0(-)
is the orientation cover of (-),
II(SM,
M) is the normal
cover of SM in M and + denotes sum of isomorphism classes of Z -
2
bundles.
In particular, if f is null-homotopic, we can take SM
=
50
and
M becomes an oriented manifold.
NowProposition 4.
1
applies.
where the obvious identifications are made in the union.
We note that an (S, q)-mock bundle does not have a total space or
a projection. But nevertheless we can set up a theory of (S, q)-mock
bundles in analogy to the usual case: The various notions of isomorphism,
restriction, cobordism etc. for (S, q)-mock bundles are obtained by
obvious modification from the definitions for ordinary mock bundles. We
leave the reader to write down the details and to establish the existence
of:
Let
a
be the category defined as follows. The objects of Ci are
linked resolutions; a map of resolutions f:
p •••p' (p, p'
ECi) is a
morphism of Ci if f
=
nf , where n is a positive integer and f is a
o
0
quasi-linked map. Clearly f and n are uniquely determined by f.
o
Let f be as above, then to each
(p,
n)-manifold M there is
associated a (P', n)-manifold f(M) defined by f(M)
=
disjoint union of
n copies of f (M).
o
An Ci-system on a ball complex K is a covariant functor
S
=
SK : ~
-+
Ci. Let
~a
be the class of S(a)-manifolds and
:m:
=
~K)
=
u ~. If K is oriented then an (S, q)-mock bundle ~q over
a a ---'----
K consists of a function ~: K
-+~,
such that, for each
a
EK,
~(a)
is
an
(S(a),
q+dim a)-manifold, labelled by
a,
with
Fig.
19
Corollary 4. 2. Suppose that the orientation cover
0
(Mm) of M
is iS0I!l0rphic to f*(1)), where 1) is the non-trivial double covering of
the circle S1 and f: M .••S1 is a map. Then there is a duality iso-
morphism,
Proof. We regard M as a Z -manifold in the following way.
2 1 -1
First we make f transverse to a point XES, then take SM
=
f (x)
o
0
and give orientations to M - SM and SM. The fact that SM is orientable (i) an abelian group nq(K, L; S), the q-th cobordism group of
follows immediately from the equation. (K, L) with coefficients in S, which depends only on S and
I
Lie
I
K
I.
SL
is an epimorphism. The injectivity of
l/I
follows from the same argu-
ments applied to P x I.
The proof of the proposition is complete.
Another example of a polyhedron, whose local homology gives rise
to a quasi-linked stack in a natural way, is provided by any unoriented
pl manifold Mm. We leave the reader to make the obvious definition
of nq(M,
£)
in this case, while we establish the following fact about M,
which is an easy consequence of Proposition 4.
1.
128 129

(ii) a spectral sequence
VII·The geometry of CW complexes
REFERENCES FOR CHAPTER VI
Another consequence of this chapter is that CW complexes,
already useful as homotopy objects, now have a beautiful intrinsic geo-
metric structure. This has strong connections with stratified sets and
the later work of Thorn, see §3. We intend to write a paper [5] examining
transverse CW complexes in greater detail and showing that they have
all the properties enjoyed by cell complexes and semisimplicial com-
plexes. In particular block bundles or mock bundles with base a trans-
verse CW complex can be defined and have good geometric properties
(see also §4of this chapter).
In this final chapter we draw together all the ideas of the previous
chapters by showing that an arbitrary cohomology theory is a geometric
.theory in an essentially unique way. Thus the geometric definitions of
coefficients, operations etc. all apply to an arbitrary theory. This is
achieved by examining the geometry of CW complexes. We will define
a new concept, that of a transverse CW complex, which has all the
geometric properties of ordinary cell complexes. In particular, it has a
dual complex and transversality constructions can be applied. The
transversality theorem (in §2) is a version for a CW complex of the
theorem in part II §4. However the proof uses even less and is ele-
mentary!
If X is a based transverse CW complex and X* its dual com-
plex, then the subcomplex
X
(X)
c
X*, consisting of dual objects other
than the object dual to the basepoint, behaves with X exactly like the
base of a Thom complex behaves with the whole complex. A map
f :
M - X can be made transverse to
X
(X) (in fact the transversality
theorem does exactly that) and the transverse map is determined by its
values near f-
I
X
(X). In this way, an arbitrary spectrum is seen to be
a 'Thom' spectrum for a suitable theory (with singularities), see §§4
and 5.
1.e. H.(X, X-x;
Q)
=
1
P
Q
for
Q
and define
does not depend on the par-
(b) X
=
rational homology manifold.
H.(Sn;
Q).
In this case fix a linked resolution
1
nq(K;
Sp )
as in (a). Again nq(K; S )
Q
P
Q
ticular resolution.
The sheaf of local homology, considered in §4for a Z -manifold,
n
provides examples of
<t
systems for many classes of polyhedra. Here
we mention three:
(a) X = homology n-manifold, i. e. H.(X, X-x; Z) = H.(Sn; Z).
1 1
Let
Pz
be any short linked resolution of Z. Then, for each
aE
K,
/KI =X, we set
S(a)=PZ
and
SPz(a<
T)=m
o
id:pZ-pZ where
m : Z - Z is the multiplication given by the local homology stack. Using
the spectral sequence (ii) as in the proof of Theorem 2. 3, we are able to
conclude that nq(K; S ) is independent of the presentation
Pz
and
Pz
therefore provides a good definition of cobordism with coefficients in the
sheaf of local n-homology of X.
(c) X is a polyhedron with only two intrinsic strata. Again
there is a good definition of cobordism with coefficients in the sheaf of
local homology. We leave details to the reader.
Finally, we conjecture that, at least when there is a 'good'
definition of cobordism with coefficients in the sheaf of local homology,
there is a 'Zeeman spectral sequence', cf. [1], relating bordism with
cobordism with local coefficients. Proposition 4. 1 gives exactly this
for a Z -manifold.
n
[1] E. C. Zeeman. Dihomology III. Proc. Cam. Phil. Soc., (3)
13 (1963), 155-83.
130 131

-1
0 "'"
A plan for Y is a map d: Y
-+
sY such that d (C(b.)) = b. and
-- 1 1
hen {Y, {b.}, d) is a building with bricks {b.} and plan d. We
1 - --- 1
hen have a new partition
some j}
st(h.)
=
b.lk(h.),
1 1 1
v
b.
lb.
=
b.
111.
n
J
b.
lb.
=
b.}
1 1 1
n
0
C*(b.)=cl{b.
b.
1 1 1
o
1
st(b.,
sY)
=
cl
{b. b.
III
o
1
n addition to the usual
The main theorem in §6is that the stable homotopy category is
equivalent to the category of geometric theories (theories of 'manifolds'
with singularities, labellings etc., see Chapter IV) with 'resolutions'
formally inverted. Thus any theory has cycles unique up to resolution
of singularities, and any natural transformation of theories is equivalent
to a relabelling followed by a 'resolution', see the examples in §6. Thus rhen if we write
a theory has products if and only if the product of two cycles in the theory
can be relabelled and then 'resolved' to give a cycle in the theory, see
v v v v. v
C(b.) = b.C(b.), C*(b.) = b.C*(b.),
also the final remark of §6.
1 1 1 1 1 1
Throughout the chapter we will use the standard notation for CW C(i;i) = C(bi) - C(bi), etc.,
i
j
e , e etc. h' . I h h'
ere IS a canomca omeomorp Ism
complexes. Thus X, Y etc. denote CW complexes, e , e ,
1 2
denote cells; all cells have given characteristic maps denoted
h : D ...•X, h : Dj
-+
X etc., where D
j
= [-
1,
l]j
C
Rj. We denote
1 1 2
h (0)
€
e
C
X by
e
in analogy with the notation for the barycentre of
1 1 1
a simplex. Other notation is in §l.
We would like to acknowledge a helpful conversation with
C. T. C. Wall at the beginning of the work of this chapter.
1.
BIDLDINGS dY (the derived of Y) = {d-1(a), (]
€
sy)
lndthe dual building
} } -1
0 .•...
y* = {Y, {b:" , d where b:"
=
d (C*(b.)).
1 1 1
In this section we describe a general structure which allows a
dual structure to be defined The examples will be used in later sections.
Let Y be a space and {b.} a partition of Y into disjoint
1
subsets. Write b.
<
b. if i
*
j
and b.
C
b..
Define the simplicial
1
1
1
1
complex sY to have for typical n-simplex a string
Notation.
,.. -1 "'"
We write b. for d (b.).
1 1
iistent.
(b.
<
b.
< ... <
b. )
1 1 1
o
1
n
C(h.)
=
cl
{b. b. '" b.
lb.
=
b. }
1 1 1 1 1 1
o
1
n n
Examples
1.1. 1.
Let K be a simplicial complex and sK
=
dK
he usual first derived of K. Then {K, {
(1),
id. } is a building with
Iricks the open simplexes and plan the identity. Our notation is then
and faces given by omitting members of the string. Write
1\
for the :onsistent with the usual notation for barycentres and first deriveds.
vertex (b.) of sY and
b. b. '" b.
for the simplex (b.
<
b.
< ...
b. ).
1 10 \
In
10 11
In 2. Suppose Y is a CW complex in which the closure of each
sY has the structure of a cone complex in the sense of McCrory [3]; :ell is a subcomplex. Let {e.} be the open cells of Y (which partition
1
in particular we can define
I)
then {Y, {e.}, d) is a building, where d is defined by inductive
1
:onical extension. Then
e.
is the centre of e. and the notation is con-
I 1
and
132 133

llso transverse.
We say that the CW complex X is transverse, or that X is a
rcw
complex, if each attaching map is transverse to the skeleton to
uhich it is mapped.
2. TRANSVERSALITY
3. Supposep : E
-+
K is a mock bundle, over a simplicial
complex or a cone complex, in which each block has a collar and the
projection is defined by mapping collar lines radially (the 'canonical
projection'). Then IE, {open blocks}, p} is a building where
sE
C
dK (the usual first derived). Note that if b. is an open block of
1
Transversality theorem 2.
1.
Suppose X is a TCW complex
E then
b.
=
cl
(b. - collar).
1 1
md f: M
-+
X a map, where M is a compact
pl
manifold. Suppose
4. Suppose X is a stratified polyhedron in the sense of Stone
I
aM is transverse. Then there is a homotopy of f reI aM to a
[61.
1.
e. X is partitioned by open manifolds, the strata, and provided ransverse map.
with a system of regular neighbourhoods. There is also a local triviality
condition which does not concern us. Then X defines a building with Corollary 2.2. Any CW complex gives rise to a TCW complex
the strata for bricks in which the plan is obtained by using the mapping Ifthe same homotopy type obtained by inductively homotoping attaching
cylinder structure of a regular neighbourhood. Then
b.
=
cl
(b. - neigh- naps to make them transverse.
1 1
bourhood of previous strata), the closed stratum. Proof of the transversality theorem. Since im(f) is contained
n a finite subcomplex of X, we can assume X is finite and proceed by
nduction on the number of cells of X. Suppose X
=
Xo
u
e
r
Transversality for CW complexes works nicely with either the Choose a collar c for aM in M and
by
a preliminary homotopy
smooth or
pi
categories. In this section we choose to work with the
pl
'el aM assume that f is constant on collar lines. Now apply the
category but a similar treatment is possible with the smooth category. ;tandard transversality theorem (see Remark 2. 3 below) to make f
Let M be a closed
pl
manifold and X a CW complex. A maPransverse to
e.
in e., by a homotopy of f reI im(e). We now have a
-1 1 1
f : M
-+
X is transverse if for each cell e.
E
X either f (e.)
=
¢
or liagram
----- 1 1
there is a commuting diagram
cl(T.)
t ~
D.
l~ . /. 1
fl ~
1/
hi
X
T
t"
D.(E)
f~/h:1
X
vhere D.
(E)
is a small disc in D. centred on
0
and t is a trivial bundle.
1 1
where T.
=
f-
1
(e.), h. is the characteristic map for
e.,
t. is the pro- 1I0reover flim(c) is already transverse to the whole of
e..
Hence, by
1 1 1 1 1 1
jection of a
pl
bundle (necessarily trivial) and
cl(T.)
has codimension :omposing flcl(M - im(c)) with a standard homotopy of
e.
in itself
All
zero in M. Notice that this implies that T.
=
t~I(O) is a submanifold obtained by expanding D.(E) onto D. and using h.) and then extending,
1 1 1 1 1
of M of codimension i. Notice also that if X
=
X
u
e. then
IY
using the collar in the usual way, to a homotopy of f which keeps
o
1
M
=
M U
T.
x D., where
T.
x D. has the form of a 'generalised handle'lM fixed, we have the same diagram with D. replacing D.(E).
o
1 1 All
A
A
-1
1
1
attached to M by T. x
an..
Nowwrite liT
=
T x aD. (T
=
t
(0)
as usual), M
=
cl
(M - T),
o
1 1 1 0
In the case that aM"*
¢,
we insist that cl (T.) meets aM in a II
=
cl (aM - T). Then M is a compact manifold with boundary
A A A
1
I
1 0
subtube T' x D. where T'
C
aT. has codimension O. Thus f aM is ~ U liT Now from definitions and the diagram both f
1M
and f
I
liT
1 1 1 • 1
134
135

are transverse to X (and to X). Thus f 13M is transverse to X
o
0 0
and by induction we may homotope f further reI 3M to make it trans-
o
verse, as required.
Corollary 2. 6. Any map between TCW complexes is homotopic
to a transverse map. The homotopy can be chosen to keep fixed a sub-
complex on which the map is already transverse.
3. FRAMIFIED SETS
Definition 3. 1. A framified set ~ of length n consists of
(1) A polyhedron X with a filtration
Roughly speaking, a framified set is a stratified set in the sense
of Stone [6] in which all the block bundles are trivialised. The precise
Proof. Use the transversality theorem inductively to shift cells
to make the map transverse. The lemma and induction ensure that a cell
is already transverse on its boundary.
The proof of Lemma 2. 5 uses the description of M as a framified
set which is given in the next section and will thus be left for convenience
until the end of that section.
Remark 2.3. We only used the simplest
pl
transversality
theorem, which has an elementary proof using the fact that the pre image
of the barycentre of a top dimensional simplex by a simplicial map is
framed - a fact that has been known for about forty years! The corres-
ponding smooth theorem is also elementary, using Sard's theorem.
Nowlet M+
=
M
U
OMx I, i. e. M with a collar glued on
3Mx 0
'on the outside', and let q : M+
-+
M be the map which projects the
collar back onto 3M. If f : M
-+
X is a map, then define f+: M+ ..• X
to be f
0
q. We say f : M
-+
X is weakly transverse if f is trans-
--~----- +
verse. Thus f 13M is transverse but some of the tubes T. might only
1
have codimension 0 in OM. An example of a weakly transverse map is definition is similar to the definition of killing in Chapter IV, and in fact
a characteristic map for a cell in a TCW complex. In fact it can easily a precise connection will be formulated in §4. The definition is by in-
be seen that X is a TCW complex if and only if all the characteristic duction on the length of filtration.
maps are weakly transverse.
A map f : X
-+
Y between TCW complexes is transverse if
f
0
h. : D.
-+
Y is weakly transverse for each characteristic map h. of
1 1 1
X.
{X
=
X :::>X :::>... X :::>X +1
=
¢}
1 2
n n
(3) For each i:S n a framified set L. of length i - l.
-1
(4) For each i:S n an isomorphism of filtered sets
'rheorem 2. 4. There is a category TCW consisting of TCW h th t
--------~-~ - suc a Xi - XH1 is a manifold for i
=
1, 2, ... , n.
complexes _<l:~!!ansverse maps. The inclusion TCW
C
CW induces (2) A regular neighbourhood system for the filtration, N.. ,
~E_~q~'yalenceof homotopy categories. TCW is closed under cross 1 :Sj :S i :Sn, (see Remark 3. 2 below).
1,
J
prodl.l~t:.L~<:dg_eproduct, factorisation of a subcomplex and smash product
!EE-a!~1:!:~~_!!.j.~c~ose~nder suspension (smash product with SI).
Proof. That TCW is a category follows from 2.5 below and
definitions. That TCW
C
CW induces an equivalence of homotopy
categories follows from 2.2 and 2.6 below. The rest is a matter of
trivial verification.
Lemma 2. 5. f: M
-+
X, g : X
-+
Y are both transverse maps
where M is a compact
pl
manifold and X, Yare TCW complexes.
Then g
0
f : M
-+
Y is transverse.
h. : {N. 1:::>N. 2 :::>... N.. } ~ N.. x C.
1 1, 1, 1,1 1,1 - 1
where C
1'
=
{C(l. 1):::>'" C(L.. 1):::>pt.} is the cone on L. with
- 1, 1,1- -1
the cone point added as the final stage of the filtration.
The cone flag C. is a fibre or model for the framified set X
-1 -- --- -'
and the framified set L. is a link for X. There are obvious notions of
-1 -- -
restriction of a framified set to a suitable subpolyhedron and of product
136
137

of a framified set with a manifold. A
pl
homeomorphism is an iso-
morphism of framified sets if it commutes with all the extra structure.
In particular two isomorphic framified sets have the same (or identi-
fiable) system of links. The final condition is:
(5) h. restricts to an isomorphism of framified sets
1
{N.
1:J
N.
2 :J ...
N..
I}
3!
N..
x
L..
1, 1, 1,1- 1,1-1
Remark 3.2. A regular neighbourhood system is constructed
inductively by defining N
==
X
and N . is a simultaneous system
n,n n n,J
of second derived neighbourhoods of
X
in
X ..
Then define
n
J
X:
==
X -
int(N .) and proceed with the construction for
1 n,1
X'
:J
X'
:J :J
X'
:J
¢
1 2 •••
n-l .
Existence and uniqueness of regular neighbourhood systems thus follows
from the usual regular neighbourhood theorem.
Notice that a framified set is a building with bricks the strata
{X. - X.
I} and plan defined by using the cone structures (see §l
1 1-
Example 4). We are not interested in the specific ordering of the strata
of ~ but only in the partial ordering given by the geometric structure
of
X,
and we will allow an isomorphism of framified sets to change the
order. The importance of framified sets lies in the following theorem,
which is essentially an observation.
Theorem 3. 3. Let f: M ...•
X
be a transverse map to a TCW,
then M has the, structure of a framified set determined up to isomorphism
by the map f.
Proof. Since the image of f is contained in a finite subcomplex
of
X
we can assume without loss that
X
is finite and proceed by in-
duction on the number of cells of
X.
In fact we will produce a framifica-
tion of the same length as the number of cells. Let
X
==
X
u e. Define
a
Mt
==
f-1(e) and Nt
1
==
cl(f-1(e» (i. e. Mt
==
of
and Nt
1
==
T in the
, ,
notation of the previous section). Now let M
==
cl (M - T) then M
a a
and IlT
3!
T
x S.
1
have the structure of framified sets by induction.
J-
138
Moreover, by uniqueness we can choose the structure on aT to be the
product of the framification of S.
1
with
T.
Choose the indeXing to
J-
agree and extend the strata to M by adjoining the 'cones' on their
intersection with IlT. L e.
M.
==
(M ).
U
T
x (C(S.
1)' -
cone pt. ).
1
a
1
J-
1
Finally the isomorphism ht is provided by the chosen product structure
on T. Induction now gives all the structure of a framified set to M.
Uniqueness is clear.
Fig. 20
We can also describe this framification (at least as a building)
quickly as follows. Take the dual complex to
X
and let
X - X*
:J
X* ::> ::>X* -
e
A
- 1 2 •••
t-
-1
be the corresponding filtration of
X.
Then Mt
==
f
(X;).
In other words
the building is the pull-back by f of the dual building to
X.
Notice that the 'models' C. are just the closures of the cone
-1
flags
139

e. n (X'" ~ X'"
::J •••
xn .
1 1 2 1
Since the models depend only on
X,
we say that the framification of M
is modelled_on
X.
It is clear that there is a 1 - 1 correspondence
between transverse maps f: M -
X
and framifications of M modelled
on
X.
This observation will be developed in the next section when we
'classify' homotopy classes of maps from one TCW to another. To end
this section we will prove the lemma left at the end of the last section.
First observe that, by inductively choosing collars on the frontiers
of the N.. we can fwd an isomorphic system with smaller closed strata
1,
l
and larger cone flags
(d.
Stone on 'minidivision'
[6]);
this has the effect,
for the framification given by a transverse map f; M ...•
X,
of replacing
• A. A
+
the nelghbourhoods T. x D. by nelghbourhoods of the form T~ x D.
A A
1 1 1 1
where T: ~ T. and D:t-
=
D.
u
collar, and it is not hard to see that we
1 1 1 1
A
o
can choose all the collars so that there are diagrams
X
We call this framification an extension of the original one.
-1
Proof of Lemma 2. 5. Let ekbe a cell in Y and Tk
=
(g
0
f) ek.
We have to show that cl(Tk) is a product Tk x Dk of codimension zero
in M which meets
aM
in a similar subproduct. Choose an extended
framification for f: M -.
X.
This means that we can regard M as
made of generalised handles of the form T ~x D:. Since go f
I :
D:
-Y
1 1 1
is transverse (from the definition of transversality for TCW's) we have
+
Din Tk
=
Q
x Dk say where
Q
is a manifold with boundary and
Q
x Dk meets
aDt
in a similar subtube. Here we are regarding
Dt
as included in M as pt. x D:
C
T~ x D:. Thus
1 1 1
6:
x D:'-) n T.
=
(Q
x T~) x Dk
1 1 1 1
=
Q'
x Dk say.
140
But all the product structures are coherent in the Dk factor and the
required product structure on T. is seen.
1
4. MANIFOLDSANDMOCKBUNDLESMODELLED ON
X
We now rephrase the results of §3by omitting the first stratum
throughout. The idea is to obtain a formulation which is invariant under
suspension and thus carries over to stable maps.
Let Xbe a based TCW. Let
*
E
Xbe the basepoint (regarded
as the first cell in
X)
and let
x
(X)
be the subcomplex of
X*
consisting
of the duals to cells other than
*.
If f : M ...•
X
is transverse then the
associated stratification of
M
starts
M:) X(M,
f) where
X(M,
f)
=
-1
f
(X
(X)).
Let e. be a typical cell of
X.
Collapse the notation of §3
1
and write C.
=
cl(x(X)
n e.), then C. is the cone e.L., where L, is
1 1 1 . 11 1
the link associated to e
1.,
and we label the cone point bye .. L. is in
1 1
fact a framified set embedded in SI'
=
aD ..
The set of basic links {L.)
1 1
defines a theory of manifolds with singularities (see IV §3) called free
X
manifolds. Using the fact that L.
C
S., we have an intuitive notion
1 1
of a framed
X
manifold. The precise formulation is in terms of
killing as in IV §4. Suppose
X
is finite and
X
=
X
U
e.. Suppose
o
1
inductively that framed
Xo
manifolds have been defined so that Liis
a framed
Xo
manifold. The theory of framed
X
manifolds is the theory
obtained from this theory by killing L. and labelling the new stratum of
1
singularities bye .. In general define a framed
X
manifold to be a
1
framed
X'
manifold where
X'
C
X
is a finite subcomplex. From the
definitions of killing and framified sets, it is easy to see that a frami-
fication of M modelled on
X
is equivalent to a framed
X
manifold
embedded in M. From Theorem 3. 3 and the transversality theorem we
have:
Proposition 4. 1. There is a 1 - 1 correspondence between the
set of homotopy classes of maps [M,
X]
and the set of cobordism classes
of framed Xmanifolds embedded in M. In particular
1T
(X) ~ Un where
n n
X--
UX means the group of cobordism classes of framed
X
manifolds em-
bedded in Sn.
141

We can extend the proposition to maps [Y, X] where Y is an
unbased TCW using an extension of mock bundles to TCW's. Let Y
be a TCW then a subset E
C
Y is the total space of an embedded mock
bundle (of dimension -q) provided that for each cell ei
E
Y there is a
diagram
where M.
=
h~1(E) and is a proper submanifold of D. of codimension
1 1 1
q. The following proposition is a generalisation of Lemma 1. 2 of Chapter
II and is proved by a similar argument to Lemma 2. 5. We omit the
details.
Proposition 4. 2. Let E
C
Y be an embedded mock bundle and
f : M
-+
Y '!-.i!'ansy~!,~emap, then f-1(E) is a proper submanifold of M
<2!...
codi}ne.!lsion q.
The proposition implies that mock bundles can be pulled back
and hence give a contravariant functor on TCW.
There is an obvious extension of the notion of mock bundle to mock
bundle with singularities and framed mock bundle. The next proposition
is, like 4. 1, essentially an observation:
Proposition 4. 3. There is a 1 - 1 correspondence between
trans~-.!'~e maps Y - X and framed X mock bundles embedded in Y.
Hence [Y, X] ~ nX(Y)' where 0X(Y) denotes cobordism classes of
frameQ X mock bundles in Y.
There is a based version of 4. 3; [Y, X]*
=
QX(Y) where 0X(Y)
denotes cobordism classes of framed X mock bundles in Y - * .
We will now stabilise these results. Let SX denote the (reduced)
suspension of X which is again a TCW then X(SX)
=
X(X) and if
f : Sn
-+
X is transverse then so is Sf: Sn+1
-+
SX. Moreover
X(Sn+1, Sf)
=
X(Sn,
f)
and the only difference between the framification
of Sn+1 given by Sf and that of Sn by f is that all the bundles are
142
enlarged by adding a trivial I-disc bundle. We thus have the stable
version of 4. 1 and 4. 3 (for details of the stable category see Adams [1]):
Proposition 4. 4. There is a 1 - 1 correspondence between stable
homotopy classes of (based) maps Sn
-+
X and cobordism classes of
stably framed X manifolds. There is a 1 - 1 correspondence between
stable homotopy classes of stable maps {Y, X) and framed X mock
00
bundles over Y - * (i. e. embedded in S Y - *).
Remarks. 1. The above notion of a mock bundle over Y is a
direct generalisation of the definition for cell complexes in Chapter
n.
2. We have been deliberately careless (or rather uninformative)
about dimensions; this will be remedied in the next section.
3. See Example 2 at the end of the next section for a clarifica-
tion of the relation between this representation and 'killing'.
5. THE CYCLES OF A HOMOLOGYTHEORY
At the end of the last section we had represented a cohomology
theory, which came from the suspension of a CW complex, as a mock
bundle theory. In this section we extend this result to arbitrary spectra
and observe that the corresponding homology theory is the corresponding
bordism theory. The results are most elegant for connected spectra
when we will be able to represent h () by n-manifolds with singularities.
n
For non-connected spectra we will need manifolds of non-constant dimen-
sion. Uniqueness of representatives will be discussed in §6.
We follow Adams' treatment of the stable category, [1].
A spectrum X is a sequence
ix.,
q., i ~
0)
of based CW com-
~---- 1 1
plexes and based maps where q. : SX.
-+
X'+1is an isomorphism onto a
1 1 1
subcomplex. It is connected if X. has no cells other than * in dimen-
---- 1
sions
<
i.
Suppose X is connected and, for some i>
0,
X.
=
*
ueiue u•..
- 1 1 2
Make X. into a TCW and, by further homotopies, ensure that each
1 .
cell e ,e ... wraps non-trivially around e
1
(geometrically, not
2 3 1
homotopically
!).
Then X(X.)
=
e
u ... will have constant codimension
1 1
143

i in X.. Now SX. is again transverse and since SX.
C
X'+l we can
1 1 1 1
make Xi+1 transverse keeping SXifixed and, by further homotopies,
ensure that each cell in X. - SX. wraps non-trivially around Se .
1+1 1 1
Thus X(X. 1) again has constant codimension (this time i + 1). Pro-
1+
ceeding in this way we can make each X. transverse, for
j
2::
i, with
l
SX.
C
X'+l and X(X.) of constant codimension j.
J J J
Nowdefine the geometric homology theory associated to ~ by
taking for basic links all the links defined by X. for j
2::
i, with the
l
obvious identifications given by the inclusions SXj
C
Xi+l' Each link
is stably framed and it makes sense to talk of stably framed ~ manifolds
and from the results of §4, Chapters II and IV and definitions we have:
Theorem 5. 1. The homology and cohomology theories defined
by the spectrum ~ are equivalent to the bordism and mock bundle
theories based on stably framed ~ manifolds. Moreover by choosing
X(X.) to have constant codimension j we have ensured that h (~;~X)
l -------------
n -
is represented by ~ manifolds of (genuine) dimension n and that
hq( ;~) is represented by mock bundles of fibre dimension q.
Remark 5.2. The first half of the theorem is true for non-
connected spectra by the same proof but since X(X.) will in general
l
have varying codimension in X., the cycles will be of mixed dimension,
l
possibly of unbounded dimension. See the examples below.
Examples 5.3. 1. X.
=
sj. X(X.)
=
pt. and an X manifold
l l -
is just a framed manifold. Thus we have the usual representation for
stable (co)homotopy.
2. For some j, Xi
=
Si
u
fDkwhere f : Sk-1 - Si is a given
map and
X.
=
S\
i
< i,
1
Z
XZ
+
i
=
S Xi'
Then
L
=
0, C
=
pt. (of codim i),
1 1
L
=
Mk-i-1C
=
C(M)
2 ' 2
144
where M is the framed submanifold corresponding to f. ~ manifolds
have two strata both framed and the neighbourhood of the smaller stratum
in the larger stratum is a product with C(M).
In other words ~ theory is framed bordism with M 'killed',
see Chapter IV §4. This example is the germ of the whole construction.
Each new cell attached determines a manifold in the previous theory,
which, when killed, gives the new theory.
3. The case f "'"
*
of the example 2 is also worth discussing
in detail. If f
=
*,
i. e. X.
=
sj
v
i<,
then the theory is a 'mixed
J
dimension' theory, i. e. an 'n-cycle' is the union of a framed n-manifold
with a framed (n+j-k).,.manifold. If f '"
*
then M is a framed manifold
bordant to
0.
Thus X theory is not a mixed dimension theory but it has
a resolvable singularity: there is an elementary resolution connecting
the two theories, see Example 6.4(4).
4. X
=
Thorn spectrum ~ or
!Y!9
etc. MPLnhas a cell
structure in which each cell is of the form (cell of BPL ) x 'in. Then if
n
BPL is a TCW, so is MPL and X(MPL)
=
BPL • Thus X(M, f)
n n n n
is just a submanifold of M of codimension n and we recover the usual
representation of bordism. Similarly with ~ we get submanifolds
with normal vector bundles, i. e. smooth(able) submanifolds. In this
example the singularities are 'virtual', they are of the form C(framed
sphere) and serve to allow the normal bundle of X(M, f) to be non-
trivial.
. '+1
5. X
=
Moore spectrum. E. g. for Z , X.
=
SJ
U
DJ.
- n
J
n
Then L
=
¢,
L
=
n points and we get framed Z manifolds. In
1 2
n
general the geometric description fits with that given in Chapters III
and IV (for stable homotopy, but see also Example 8 below).
6. If ~ and yare TCW spectra then so is ~
v
y and an
X
v
Y manifold is merely the union of an ~ manifold with a y mani-
fold. Take care about dimensions, see Example 3 above.
7. If ~ and
¥
are TCW spectra then so is a naive smash
product ~
A
y, see [1; p. 161], and a cell of ~
A
Y
is the product of one
of ~ with one of y and has for link the ioin of the corresponding links
145

for X and Y. Thus an X
A
Y manifold is a manifold with singularities
•....
"",
.
being joins of ~ singularities and
y
singularities.
8. Combine Examples 4, 5 and 7. Then bordism with co-
efficients has exactly the description given in part III!
6. RESOLUTIONOF SINGULARITIESANDTHE MAIN THEOREM
Let T be a geometric homology theory and suppose that in T
there are two singularities of the form
(1) C(M)
(2) C(C(M)
U
MW) where W is a bordism (in T) of M to
fi.
Suppose also that C(M) does not appear in any other basic link.
Then a T-manifold can be resolved so as to delete both of these two singu-
larities. This is done by resolving all the C(M) type singularity using
(2) which is essentially a bordism in T of C(M) to a T-manifold with
no C(M) singularity. The method is similar to the proof of exactness
in IV Proposition 4. 1. Once there are no C(M) singularities, then
there can be none of the second type either. The precise description of
this resolution process is contained in the CW interpretation which
follows. Let T' be the theory in which these two singularities do not
appear then we say that T' is an elementary resolution of T. A
simultaneous family of elementary resolutions will also be called an
elementary resolution. A resolution is a countable sequence of elementary
resolutions.
Nowsuppose that X and X' are TCW's and T and T' above
are the corresponding theories. Then X'
C
X and X differs from X'
n n+1. n+1 b 1
by the addition of two cells e and e wIth e attached y degree
on en and en otherwise free. In other words X' differs from X by
an elementary collapse. Thus the geometric analogue of collapsing is
resolution and the process of resolution of a given T-manifold described
at the beginning of the section is just transversely deforming the map
Sn
-+
X (which defines the manifold) into X', using the deformation re-
traction X
-+
X'. The inclusion X'
C
X is an elementary expansion.
An expansion is a countable sequence of (simultaneous families of) ele-
mentary expansions.
146
The following theorem is proved by an argument similar to that
contained in Chapter I (in fact rather simpler).
*
Its proof will therefore
be omitted.
Theorem 6. 1. Let CW and SCW denote the categories of CW
complexes and isomorphisms onto subcomplexes and CW spectra and
isomorphisms onto subspectra respectively. Let
L
denote the expansion
in CW or SCW. Then there are isomorphisms
hCW
£:;<
CW(L-
1)
hSCW
£:;<
SCW(L-
1).
Combining with 2.4 we have:
Corollary 6. 2. There are equivalences of categories
hCW "" TCW(L-
1)
hSCW "" STCW(L-
1).
Nowlet
9
denote the category whose objects are geometric
theories and whose morphisms are inclusions of theories or 'relabellings'.
We now regard all theories as theories of framed manifolds with singu-
larities, non-framed manifolds being dealt with as in Example 5.3(4), by
allowing singularities corresponding to the twisting of the normal bundle.
A theory T is included in a theory T' if the links of T' include those
of T up to a relabelling. Let R denote the resolutions in g. From the
discussion at the beginning of this section and 5. 1 we have an isomorphism
Combining this with 6.2 we have:
Main theorem 6. 3. There is an equivalence of categories
g(R) "" hSCW .
*
There are similar set theoretic problems to those encountered in
Chapter I §4. These can be dealt with in a similar way.
147

is an operation and in fact many classical operations have this form,
(cf. McCrory
(4)).
ties.
REFERENCES FOR CHAPTER
vn
Remarks and corollaries 6. 4. 1. There is an unstable version,
using 'embedded' geometric theories (corresponding to the embedded
mock theories of
§4).
We obtain an equivalence ES (R) "" hCW.
In other words the stable homotopy category is equivalent to the category
of geometric_t!!f!0ries with resolutions_ formally inverted. The operations
v
and
A
in hSCW are described geometrically as 'union' and 'join'
of singularities, see 5.3(6) and (7). 6. Finally a theory with products is one in which there is a
map ~
A ~ ••••~
with suitable properties. By expanding ~ by a sequence
of mapping cylinders we can replace this by an equivalent inclusion
~' A
~'C ~'.
Now if ~n ...•~', ~m ...•~' represents two manifolds in
the theory then §n
A
§m ...•~'
A~'
represents their geometric prod-
uct (see 5. 3 Example 7). Thus a theory has products if and only if it
2. Theorem 6. 3 gives the answer to the problem of uniqueness is possible to find a geometric representation in which the product of two
of geometrl·c representatl·ves for a given homology theory. Two geometric
0 0
o.
0
cycles IS agam a cycle (after possIbly relabellmg). Thus the classIcal
theorl·es glOveequl·valent homology theories if and only if they differ by a
0 0 0 0
examples of rmg the ones (bordlsm etc. ) are essentIally the general
sequence of resolutions and their inverses. example! Note however the case of R-bordism see Chapter III in
.
"
,
,
which the ring structure appears naturally via a resolution of singular i-
3. The theorem also describes the stable maps geometrically
(i. e. natural transformations of theories, operations etc.). Such a map
always has the form inclusion (relabelling) followed by resolution. This
follows from an analogue of Lemma 2. 1 of Chapter I. See also the next
4. The example described in 5. 3(3) fits into the setting of this
o
k
0
section as follows. Let X'
=
SJ
U
H(D x I) where H IS the homotopy [2]
of f to
*.
Make X' transverse relative to the two 'ends'. Then X'
defines a geometric theory of which both the theories described in 5.3(3) [3]
are elementary resolutions. The new singularity in X' is [4]
C(C(M)
U
a
W
U
pt. ) and can be used to resolve either the lower dimensional[5]
piece into a singularity of the higher dimensional piece, or conversely.
This example makes it clear how resolutions can change the appearance [6]
(and dimension) of a manifold drastically.
two examples.
[1]
J.
F. Adams. Stable homotopy and generalised homology.
ChicagoU.P. (1974).
P. Gabriel and M. Zisman. Homotopy theory and the calculus of
fractions. Springer-Verlag, Berlin (1967).
C. McCrory. Cone complexes and duality, (to appear).
C. McCrory. Geometric homology operations, (to appear).
C. P. Rourke and B.
J.
Sanderson. CW complexes as geometric
objects, (to appear).
D. A. Stone. Stratified polyhedra. Springer-Verlag lecture notes
No. 252.
5. If we now consider the map
sj
l,
Dk
-+
Sk
f
given by collapsing sj to a point, then the geometric description of this
map is 'restrict to the singularity'. This is seen by including sj ufDk
in Dj
+
1
U
Dkwhich collapses to ~ and using an argument like
f
Example 4. This example makes it clear that 'restriction to a singularity'
148
149
IIZ_!!!r!!l1I!ll!!,_m..
~.1iIIIII0'1III!.-.!If" -. ----------:-; , .-:~- ••••••
w_-
-Jt_