A simple homotopytheoretical proof of the Sullivan conjecture
ABSTRACT We give a new proof, using comparatively simple techniques, of the Sullivan
conjecture: the space of pointed maps from the classifying space of the cyclic
group of order $p$ to any finitedimensional CW complex $K$ is contractible.
 Citations (14)
 Cited In (0)

Article: Hspaces with torsion
Memoirs of the American Mathematical Society 01/1979; 22(223). · 1.82 Impact Factor 
Article: Reduced product spaces
Annals of Mathematics · 3.03 Impact Factor  SourceAvailable from: sciencedirect.com[Show abstract] [Hide abstract]
ABSTRACT: Let G be a tower of finitely generated nilpotent groups and let G(p) denote its localization at a prime p. It is shown that if lim1 (G) is nontrivial, then it contains at least two elements whose images are equal in lim1 (G(p)) for every prime p. Are there, in fact, infinitely many elements with this property? For abelian towers, the answer is yes. It is shown that the answer is also yes for certain nonabelian towers. Other questions considered in this paper include the problem of characterizing the lim1 term of towers of countable abelian groups, towers whose lim1 terms are countably infinite, and the extent to which the lim1 term depends upon the actual group structures in the tower.Journal of Pure and Applied Algebra 01/1995; · 0.53 Impact Factor
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arXiv:1105.3951v1 [math.AT] 19 May 2011
A SIMPLE HOMOTOPYTHEORETICAL PROOF OF THE
SULLIVAN CONJECTURE
JEFFREY STROM
Abstract. We give a new proof, using comparatively simple tech
niques, of the Sullivan conjecture: map∗(BZ/p,K) ∼ ∗ for every finite
dimensional CW complex K.
Introduction
Haynes Miller proved the Sullivan conjecture in [15]. Another proof can
be deduced from the extension due to Lannes [10], whose proof depends on
the insights into unstable modules and algebras afforded by the T functor.
There are three main problems with these proofs: they are very complicated;
the content is almost entirely encoded in pure algebra; and it is difficult to
tease out the fundamental properties of BZ/p that make the proofs work.
Our purpose in this paper is to offer a new proof avoiding these com
plaints. The Sullivan conjecture is an easy consequence of the following
main theorem.
Theorem 1. Let X be a CW complex of finite type, and assume that
? H∗(X;Z[1
(1)? H∗(X;Fp) is a reduced unstable Apmodule and? H∗(X;Fp)⊗J(n) is
an injective unstable Apmodule for all n ≥ 0.
(2) Exts
(3) map∗(X,S2m+1) ∼ ∗ for all sufficiently large m.
(4) map∗(X,K) ∼ ∗ for all simplyconnected finite complexes K.
(5) map∗(X,?∞
(6) map∗(X,K) ∼ ∗ for all simplyconnected finitedimensional CW
complexes K.
(7) map∗(X,K) ∼ ∗ for all simplyconnected CW complexes K with
clW(K) < ∞.
Furthermore, if π1(X) has no perfect quotient groups (that is, if π1(X) is
hypoabelian), then the simplyconnected hypotheses on K are not needed.
p]) = 0. Then each of the following conditions implies the next.
U(Σ2m+1Fp,? H∗(Σs+tX)) = 0 for every s,t ≥ 0 and all m ≥ 0.
i=1Sni) ∼ ∗ for any countable set {ni} with each ni> 1.
The notation clW(K) in part (7) denotes the cone length of K with respect
to the collection W of all wedges of spheres; see [17, §1] for a brief overview
2010 Mathematics Subject Classification. Primary 55S37, 55R35; Secondary 55S10.
Key words and phrases. Sullivan conjecture, homotopy limit, cone length, phantom
map, MasseyPeterson tower, T functor, Steenrod algebra, unstable module.
1
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2 J. Strom
of the main properties of cone length. Since (7) implies (3), the last five
statements are actually equivalent.
Every finitedimensional CW complex K of course has finite cone length
with respect to W. The extension from part (6) to part (7) is a geometric
parallel to the passage from finitedimensional spaces in [15] to spaces with
locally finite cohomology in [10].
Since it is known that the reduced cohomology? H∗(BZ/p;Fp) is reduced
[16, Lem. 2.6.5] and that? H∗(BZ/p;Fp) ⊗ J(n) is an injective unstable Ap
module for all n ≥ 0 [16, Thm. 3.1.1], we obtain the Sullivan conjecture as
an immediate consequence.
Corollary 2 (Miller). If clW(K) < ∞ (and, in particular, if K is finite
dimensional), then map∗(BZ/p,K) ∼ ∗.
The claim that our proof is simple should be justified: we make no use
whatsoever of spectral sequences, except implicitly in making use of the
wellknown cohomology of EilenbergMac Lane spaces; the existence of un
stable algebras over the Steenrod algebra is mentioned only to make sense of
MasseyPeterson towers; finally, the only homological algebra in this paper
is the usual abelian kind. It has seemed appropriate at points to emphasize
the simplicity of the present approach by laying out some results that might
just as well have been cited from their sources.
Acknowledgement. Many thanks are due to John Harper for pointing out
that I had the raw materials for a proof of the full Sullivan conjecture, and
for bringing the paper [14] to my attention.
1. Preliminaries
We begin by reviewing some preliminary material on the category U of
unstable Apalgebras, MasseyPeterson towers and phantom maps.
1.1. Unstable Modules over the Steenrod Algebra. The cohomology
functor H∗(?;Fp) takes its values in the category U of unstable modules and
their homomorphisms. An unstable module over the Steenrod algebra Ap
is a graded Apmodule M satisfying PI(x) = 0 if e(I) > x, where e(I) is
the excess of I and x is the degree of x ∈ M. We begin with some basic
algebra of unstable modules, all of which is (at least implicitly) in [16].
Suspension of Modules. An unstable module M ∈ U has a suspension
ΣM ∈ U given by (ΣM)n= Mn−1. The functor Σ : U → U has a left
adjoint Ω and a right adjoint?Σ. A module M is called reduced if?ΣM = 0.
Projective and Injective Unstable Modules. In the category U, there
are free modules F(n) = Ap/E(n), where E(n) is the smallest left ideal
containing all Steenrod powers PIwith excess e(I) > n. It is easy to see
that the assignment f ?→ f([1]) defines natural isomorphisms
HomU(F(n),M)
∼=
− → Mn.
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A simple homotopytheoretical proof of the Sullivan conjecture3
This property defines F(n) up to natural isomorphism, and shows that F(n)
deserves to be called a free module on a single generator of dimension n.
More generally, the free module on a set X = {xα} with xα = nαis (up to
isomorphism) the sum?F(nα) (see [16, §1.6] for details).
A graded Fpvector space M is of finite type if dimFp(Mk) < ∞ for each
k. Since Apis of finite type, so is F(n).
The functor which takes M ∈ U and returns the dual Fpvector space
(Mn)∗is representable: there is a module J(n) ∈ U and a natural isomor
phism
HomU(M,J(n))− → HomFp(Mn,Fp).
Since finite sums of vector spaces are also finite products, these functors are
exact, so the module J(n) is an injective object in U.
∼=
The Functor τ. In [16, Thm 3.2.1] it is shown that for any module H ∈ U,
the functor H⊗Ap? has a left adjoint, denoted (? : H)U. Fix a module H (to
stand in for? H∗(X;Fp)) and write τ for the functor (? : H)U; this is intended
to evoke the standard notation T for the special case H =? H∗(BZ/p;Fp).
Lemma 3. Let H ∈ U be a reduced unstable module of finite type and
suppose that H ⊗ J(n) is injective in U for every n ≥ 0. Then
(a) τ exact,
(b) τ commutes with suspension,
(c) if M is free and of finite type, then so is τ(M), and
(d) if H0= 0, then τ(M) = 0 for any finite module M ∈ U.
Proof. These results are covered in Sections 3.2 and 3.3 of [16]. Specifically,
parts (a) and (b) are proved as in [16, Thm. 3.2.2 & Prop. 3.3.4]. Parts (c)
and (d) may be proved following [16, Lem. 3.3.1 & Prop. 3.3.6], but since
there are some changes needed, we prove those parts here.
Write dk= dimFp(Hk); then there are natural isomorphisms
∼=
HomU(F(n),H ⊗ M)
∼=
HomU
HomU(τ(F(n)),M)
??
i+j=nF(i)⊕dj,M
?
,
proving (c) in the case of a free module on one generator. Since τ is a left
adjoint, it commutes with colimits (and sums in particular), we derive the
full statement of (c).
If H0= 0, then d0 = 0 and τ(F(n)) is a sum of free modules F(k)
with k < n. Since F(0) = Fp, we see that τ(Fp) = 0; then (a), together
with the fact that τ commutes with colimits, implies that τ(M) = 0 for all
trivial modules M. Finally, any finite module M has filtration all of whose
subquotients are trivial, and (d) follows.
?
1.2. MasseyPeterson Towers. Cohomology of spaces has more structure
than just that of an unstable Apmodule. It has a cup product which makes
H∗(X;Fp) into an unstable algebra over Ap. The category of unstable
algebras is denoted K.
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4 J. Strom
The forgetful functor K → U has a left adjoint U : U → K. A space X
is said to have very nice cohomology if H∗(X)∼= U(M) for some unstable
module M of finite type.
Since U(F(n))∼= H∗(K(Z/p,n)), there is a contravariant functor K which
carries a free module F to a generalized EilenbergMac Lane space (usually
abbreviated GEM) K(F) such that H∗(K(F))∼= U(F). If F is free, then
so is ΩF, and K(ΩF) ≃ ΩK(F).
Lemma 4. For any X, [X,K(F)]∼= HomU(F,? H∗(X)).
It is shown in [8,11,14] that if H∗(Y )∼= U(M) and P∗→ M → 0 is a free
resolution in U, then Y has a MasseyPeterson tower
···
??Ys
??
??
Ys−1
??
??
···
??Y1
??
??
Y0
??
K(ΩsPs+1)
K(Ωs−1Ps)
K(ΩP2)K(P1)
in which
(1) Y0= K(P0),
(2) each homotopy group πk(Ys) is a finite pgroup,
(3) the limit of the tower is the completion Y∧
(4) each sequence Ys→ Ys−1→ K(Ωs−1Ps) is a fiber sequence, and
(5) the compositions ΩK(Ωs−1Ps) −→ Ys−→ K(ΩsPs+1) can be nat
urally identified with K(Ωsds+1), where ds+1 : Ps+1 → Ps is the
differential in the given free resolution.
p,
1.3. Phantom Maps. A phantom map is a map f : X → Y from a CW
complex X such that the restriction fXnof f to the nskeleton is trivial for
each n. We write Ph(X,Y ) ⊆ [X,Y ] for the set of pointed homotopy classes
of phantom maps from X to Y . See [12] for an excellent survey on phantom
maps.
If X is the homotopy colimit of a telescope diagram ··· → X(n)→
X(n+1)→ ···, then there is a short exact sequence of pointed sets
∗ → lim1[ΣX(n),Y ] −→ [X,Y ] −→ lim[X(n),Y ] → ∗,
and dually, if Y is the homotopy limit of a tower ··· ← Y(n)← Y(n+1)← ···,
then there is a short exact sequence
∗ → lim1[X,ΩY(n)] −→ [X,Y ] −→ lim[X,Y(n)] → ∗.
In the particular case of the expression of a CW complex X as the homotopy
colimit of its skeleta or of a space Y as the homotopy limit of its Postnikov
system, the kernels are the phantom sets.
We will be interested in showing that all phantom maps are trivial. One
useful criterion is that if G is a tower of compact Hausdorff topological
spaces, then lim1G = ∗ (see [12, Prop. 4.3]). This is used to prove the
following lemma.
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A simple homotopytheoretical proof of the Sullivan conjecture5
Lemma 5. Let ··· ← Ys← Ys+1← ··· be a tower of spaces such that each
homotopy group πk(Ys) is finite. If Z is of finite type, then lim1[Z,ΩYs] = ∗.
Proof. The homotopy sets [Zn,ΩjYs] are finite, and we give them the discrete
topology, resulting in towers of compact groups and continuous homomor
phisms. Fixing s and lettting n vary, we find that lim1[Zn,Ω2Ys] = ∗, and
hence the exact sequence
0 → lim1n[Zn,Ω2Ys] −→ [Z,ΩYs] −→ limn[Zn,ΩYs] → 1
(of groups) reduces to an isomorphism [Z,ΩYs]∼= limn[Zn,ΩYs].
[Z,ΩYs] is an inverse limit of finite discrete spaces, it is compact and Haus
dorff; and since the structure maps Ys→ Ys−1induce maps of the towers
that define the topology, the induced maps [Z,ΩYs] → [Z,ΩYs−1] are con
tinuous. Thus lim1[Z,ΩYs] = ∗.
Since
?
The MittagLeffler condition is another useful criterion for the vanishing
of lim1. A tower of groups ··· ← Gn ← Gn+1 ← ··· is MittagLeffler
if there is a function κ : N → N such that for each n Im(Gn+k→ Gn) =
Im(Gn+κ(n)→ Gn) ⊆ Gnfor every k ≥ κ(n).
Proposition 6. Let ··· ← Gn← Gn+1← ··· be a tower of groups.
(a) If the tower is MittagLeffler, then lim1Gn= ∗.
(b) If each Gnis a countable group, then the converse holds: if lim1Gn= ∗,
then the tower is MittagLeffler [12, Thm. 4.4].
Importantly, the MittagLeffler condition does not refer to the algebraic
structure of the groups Gn. This observation plays a key role in the following
result (cf. [13, §3]).
Proposition 7. Let X be a CW complex of finite type, and let Y1and Y2
be countable CW complexes with ΩY1≃ ΩY2. Then Ph(X,Y1) = ∗ if and
only if Ph(X,Y2) = ∗.
Proof. The homotopy equivalence ΩY1 ≃ ΩY2 gives levelwise bijections
{[ΣXn,Y1]}∼= {[Xn,ΩY1]}∼= {[Xn,ΩY2]}∼= {[ΣXn,Y2]} of towers of sets.
Since X is of finite type and Y1,Y2are countable CW complexes, these tow
ers are towers of countable groups. Now the triviality of the first phantom
set implies that the first tower is MittagLeffler; but then all four towers
must be MittagLeffler, and the result follows.
?
2. (1) implies (2)
Write H =? H∗(X;Fp); thus H ∈ U is a reduced module of finite type and
H ⊗ J(n) is injective for all n. If P∗→ M → 0 is a free resolution of M in