Page 1
arXiv:math/0501304v2 [math.AT] 30 Nov 2007
THE UNSTABLE INTEGRAL HOMOLOGY OF THE
MAPPING CLASS GROUPS OF A SURFACE WITH
BOUNDARY.
V´ERONIQUE GODIN
Abstract. We construct a graph complex calculating the integral ho-
mology of the bordered mapping class groups. We compute the ho-
mology of the bordered mapping class groups of the surfaces S1,1, S1,2
and S2,1. Using the circle action on this graph complex, we build a
double complex and a spectral sequence converging to the homology of
the unbordered mapping class groups. We compute the homology of the
punctured mapping class groups associated to the surfaces S1,1and S2,1.
Finally, we use Miller’s operad to get the first Kudo-Araki and Browder
operations on our graph complex. We also consider an unstable version
of the higher Kudo-Araki-Dyer-Lashoff operations.
1. Introduction
Let S = Sg,nbe a surface of genus g with n boundary components. We
assume that the boundary of S is not empty. The bordered mapping class
group of S
M(S;∂) = π0Diff+(S;∂S)
is the group of isotopy classes of self-diffeomorphisms which fix the boundary
pointwise. The punctured mapping class group of S
M(S) = π0Diff+(S,∂1S,...∂nS)
is the group of isotopy classes of orientation-preserving diffeomorphisms
which restrict to diffeomorphisms of each boundary components.
These mapping class groups act on the appropriate Teichm¨ uller spaces
with quotient the moduli space of complex surfaces of the suitable type. The
action of M(S;∂) is free and the integral homology of the group M(S;∂) is
the integral homology of the moduli space of conformal structure on S with
cylindrical ends. Since the action of M(Sg,n) has only finite isotropy, the
rational homology of the group M(Sg,n) coincide with the rational homology
of the moduli space of conformal structure on a closed surface Sg,0with n
marked points.
By gluing a twice-punctured torus to the unique boundary of a surface
Sg,1with genus g, we get group homomorphisms
··· −→ M(Sg,1;∂)
ψg
−→ M(Sg+1,1;∂)
ψg+1
−→ M(Sg+2,1;∂) −→ ···
Date: February 1, 2008.
1
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2V´ERONIQUE GODIN
whose direct limit we denote by M∞. Harer [14] showed that the ψg’s in-
increasing with the genus g. In this stable range, the homology and the
cohomology groups
duce cohomological and homological isomorphisms in a range of dimensions
Hk(M(Sg,1;∂))∼= Hk(M∞)
are independent of the genus and are called the stable homology and co-
homology of the bordered mapping class group. Mumford conjectured in
[22] that the stable rational cohomology of these mapping class groups is a
polynomial algebra
H∗(M∞;Q)∼= Q[κ1,κ2,...].
in the tautological classes which are obtained from the Chern class of the
vertical tangent bundles of the universal surface bundle.
By gluing surfaces Sgi,1to the first n boundary components of the gen-
eralized pair of pants S0,n+1, Miller defined homomorphisms
Hk(M(Sg,1;∂))∼= Hk(M∞)
g ≥ 2k + 1
(1.1)
M(Sg1,1;∂) × ·M(Sgn,1;∂) × M(S0,n;∂) −→ M(S?gi,1;∂).
X =
BM(Sg,1;∂)
Using a recognition principle, he showed that the group completion of
?
is a two-fold loop space. In [28], Tillmann used a cobordism category to
extend Miller’s result. She showed that the group completion of X has the
cohomology of an infinite loop space. Madsen and Weiss [19] then identified
this infinite loop space to be Ω∞CP∞
Mumford conjecture. However, it also gives information about the torsion
of the stable cohomology. Using the Madsen-Weiss theorem, Galatius in [11]
computed the mod-p cohomology of the infinite loop space Ω∞CP∞
covering a rich and unexpected torsion component of the stable cohomology
of the mapping class groups.
These recent exciting results describe the stable cohomology of the map-
ping class groups. However, not much is known about their unstable coho-
mology. The goal of this paper is to study this integral unstable homology.
Although the unstable homology is interesting in itself, we have a specific
application in mind. In [6], Cohen and the author have used a special type
of fat graphs to define operations on the free loop space LM of an orientable
manifold extending earlier work of Chas and Sullivan [4]. It is conjectured
that these operations extend to operations
−1. Rationally, this result proves the
−1, un-
H∗(M(Sp+q;∂)) ⊗ H∗(LM)⊗p−→ H∗(LM)⊗q
particular, this would give that the Chas and Sullivan product is part of an
E∞-structure on H∗(LM). Antonio Ram´ ırez and the author are working on
homology classes found in this paper will give example of these operations on
parameterized by the homology of the bordered mapping class groups. In
using the model introduced in this paper to define these operations. Also the
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THE UNSTABLE INTEGRAL HOMOLOGY OF THE MAPPING CLASS GROUPS3
H∗(LM) and may prove essential to show the non-triviality of these higher
homological operations.
1.1. Fat graphs and the mapping class groups. A fat graph or ribbon
graph is a finite connected graph with a cyclic ordering of the half-edges
incident to each vertex. From a fat graph, we construct a surface by replacing
each edge by a thin ribbon and by gluing these ribbons at the vertices
according to the cyclic orderings.
Following ideas of Thurston, Strebel [27], Bowditch and Epstein [3] and
Penner [24] constructed a triangulation of the decorated Teichm¨ uller space
of a punctured Riemann surface S which is equivariant under the action
of the unbordered mapping class group. The quotient space, in which a
point is an isomorphism class of metric fat graphs, gives a model for the
corresponding decorated moduli space.
The spaces of fat graphs are filtered by the combinatorics of the graphs.
This stratification has been used by Penner [24], Harer and Zagier [12] to
compute the Euler characteristic of the moduli space and by Kontsevich
to prove Witten’s conjecture about the intersection numbers of the Miller-
Morita-Mumford classes in the Deligne-Mumford compactification of the
moduli space of punctured surfaces [18].
The spectral sequence associated to the combinatorial filtration collapses
rationally to a complex. This complex is generated freely by isomorphism
classes of oriented fat graphs and its boundary maps can be described com-
binatorially. Penner [25] was first to build this graph complex GC∗which
by construction computes the rational cohomology of the moduli space of
marked Riemann surfaces.
1.2. Results. To study bordered mapping class group, we first extend the
notion of fat graph. In section 2.3, we define a bordered fat graph to be a
fat graph with exactly one leaf (vertex with a single edge attached to it) for
each boundary component. Each of these leaves gives a marked point on
the corresponding boundary component. We denote by Fatbthe category
of isomorphism classes of bordered fat graphs. Using the work of Harer [13],
we show that there is an homotopy equivalence
?
between the geometric realization of Fatbwith an added base point and the
classifying spaces of the bordered mapping class groups.
Following Penner, we define, in section 3, a combinatorial filtration on
our categorical model Fatb. We show in theorem 4 that this filtration is
the skeleton of a CW-structure on |Fatb| with exactly one cell for each iso-
morphism class of bordered fat graphs. The cellular chain complex BGC∗
of this CW-structure gives the equivalent of the graph complex for bor-
dered fat graphs. Note that although the original graph complex gives an
(|Fatb|)+≃
g≥0,n≥1
BM(Sg,n;∂)
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4 V´ERONIQUE GODIN
isomorphism only on the rational homology, we get an integral result
H∗(BGC∗;Z)∼=
?
H∗(M(S;∂);Z).
Using BGC∗, we compute the homology of M(S1,1;∂) directly.
H∗(M(S1,1;∂))∼=
?
Z
∗ = 0,1
∗ ≥ 20
.
Using a computer algebra, we get
H∗(M(S1,2;∂))∼=
Z
∗ = 0
∗ = 1
∗ = 2
∗ = 3
∗ ≥ 4
Z ⊕ Z
Z/2 ⊕ Z
Z/2
0
H∗(M(S2,1;∂))∼=
Z
∗ = 0
∗ = 1
∗ = 2
∗ = 3
∗ = 4
∗ ≥ 5.
Z/10
Z/2
Z/2 ⊕ Z
Z/6
0
The results for M(S1,1;∂) and M(S2,1;∂) match the computations of Ehren-
fried in [9].
To get at the integral homology of the punctured mapping class group of
a surface S = Sg,n, we consider the exact sequence of topological groups
0 −→ Diff+(S;∂) −→ Diff+(S,∂1S,...,∂nS) −→
0 −→ Zn−→ M(S;∂) −→ M(S) −→ 0.
around the boundary components. Since these Dehn twists are central, the
bordered mapping class group is a central extension of the punctured one.
For a surface with a single boundary component, this extension gives a
double complex structure on the vector space BGC∗⊗Z[u] which calculates
the integral homology of the unbordered mapping class groups M(Sg,n).
Using this double complex and its associated spectral sequence, we compute
the following homology groups.
Z/2 ⊕ Z/6
?
Diff+(∂iS,∂iS) −→ 0
whose homotopy long exact sequence gives
In fact, the group M(S) is obtained from M(S;∂) by killing the Dehn twists
H∗(M(S1,1);Z)
∼=
Z
∗ = 0
∗ = 2k + 1
∗ = 2k + 2.
Z/12
0
H∗(M(S2,1);Z)
∼=
Z
∗ = 0
∗ = 1
∗ = 2
∗ = 2k + 3
∗ = 2k + 4
Z/10
Z/2 ⊕ Z
Z/2 ⊕ Z/120 ⊕ Z/10
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THE UNSTABLE INTEGRAL HOMOLOGY OF THE MAPPING CLASS GROUPS5
We then translate Miller’s homomorphism to our models. We get a prod-
uct on the bordered graph complex
BGCp⊗ BGCq−→ BGCp+q.
We know that this product is homotopy commutative. Using this homotopy,
we define maps
BGCp−→ BGC2p+1
BGCp⊗ BGCq
φ
−→ BGCp+q+1
which induce the first Araki-Kudo and Browder operations at the homology
level. The Browder operations are obstruction to an n-loop space being an
higher loop space. Since the infinite loop space of Tillmann extends Miller’s
double loop space, the Browder operation hit only unstable classes. The
result of Tillmann also gives higher Araki-Kudo-Dyer-Lashoff operations
Qi,p: Hk(M∞;Z/p) −→ Hpk+i(M∞;Z/p).
?Qi,p: Hk(M(Sg,1;∂);Z/p) −→ Hpk+i(M(Spg,1;Z/p).
group whose homology is a direct summand of the stable homology of the
unbordered mapping class group.
Using an idea of Cohen and Tillmann [5], we build operations
The Qi,pare obtained from projecting to the stable bordered mapping class
1.3. Remark. There exists other models for the classifying spaces of the
bordered mapping class groups and some computations in low genus have
already been made. In [13] Harer extended the notion of arc complexes of
Strebel to bordered surfaces. This model was subsequently used by Kauf-
mann, Livernet and Penner [17] to define an operad structure on a compact-
ification of the moduli space of bordered Riemann surfaces. Our proof that
Fatbrealizes to a classifying space for the bordered mapping class group
relies heavily on the work of Harer.
In [1] B¨ odigheimer has constructed a configuration space model Rad con-
sisting of pairs of radial slits on annuli. Using this model, Ehrenfried in [9]
calculated the integral homology of H∗(M(S1,1;∂)) and H∗(M(S2,1;∂). His
computations agree with ours.
Although the model introduced in this paper is close to the arc complex
model, both its categorical nature and its use of fat graphs will prove es-
sential for future applications to string topology. A category can be studied
by using techniques of algebraic topology and homotopy theory and future
applications will utilize homotopy limits, techniques of algebraic K-theory,
symmetric monoidal categories and infinite loop spaces, as well as theorems
of McDuff, Segal, Quillen and Grothendiek.
The author would like to thank Ralph Cohen, Daniel Ford, Tyler Lawson,
Antonio Ram´ ırez and Ralph Kaufmann for interesting conversations on these
topics.
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6 V´ERONIQUE GODIN
2. A categorical model for the bordered mapping class group.
Let S = Sg,nbe a surface of genus g with n boundary components. Again
M(S;∂) = π0Diff+(S;∂)
will be the group of isotopy classes of orientation-preserving diffeomorphisms
of S that fixes the boundary pointwise. In this section, we construct a cat-
egory Fatbwhose objects are isomorphism classes of bordered fat graphs.
The geometric realization of this category is a classifying space for the bor-
dered mapping class groups. More precisely,
?
2.1. Fat graphs and punctured Riemann surfaces. We briefly intro-
duce the classical fat-graph model for the punctured mapping class groups.
|Fatb| ≃
g≥0,n≥1
BM(Sg,n;∂)(g,n) ?= (0,1).
Definition 1. A combinatorial graph G = (V,H,s,i) consists of a set
of vertices V, a set of half-edges H, a map s : H → V and an involution
The map s sends an half-edge to its source. The involution i pairs an
half-edge with its other half and an edge of G is an orbit of the involution i.
The geometric realization of G is a CW-complex |G| with vertices V, 1-cells
|G|1= H/i and no higher cells.
A tree T = (VT,HT) of G is a subgraph of G whose geometric realization
|T| is contractible. A forest F of G is a subset of the set of half-edges of
G that is closed under iGsuch that the realization |F| of the combinatorial
graph
(VG,F,i|F,s|F)
is a disjoint union of contractible spaces.
We extend the maps i and s to be the identity on vertices and define a
morphism of combinatorial graph φ : G ?−→?G to be a map of sets
that commutes with s and i and such that
(1) φ−1(v) is a tree for every vertex v ∈ VeG
(2) φ−1(A) contains a single half-edge of G for every half-edge A of?G
homotopy equivalence on the geometric realizations.
i : H → H without fixed points.
φ : VG∐ HG?−→ VeG∐ HeG
Such a morphism of combinatorial graph induces a simplicial and surjective
Definition 2. A fat graph Γ is a combinatorial graph G together with a
cyclic ordering σvof the half-edges incident to each vertex v.
Since each half-edge is incident to one vertex, the cycles σvgive a permu-
tation σ = (σv) on the set of half-edges, which we call the fat structure. It
sends an half-edge to its successor in the cyclic ordering at its source vertex.
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THE UNSTABLE INTEGRAL HOMOLOGY OF THE MAPPING CLASS GROUPS7
u
A
B
C
B
A
C
v
_
_
_
(a) Γ
(b) ΣΓ
u
A
B
C
A
C
B
v
_
_
_
(c)eΓ
(d) ΣeΓ
Figure 1. Fat graphs and their associated surfaces
Example 1. The combinatorial graph of figure 1(a) has a fat graph structure
σu= (ABC)
σv= (ABC)
σ = (ABC)(ABC).
given by the clockwise orientation of the plane.
A fat graph Γ = (G,σ) thickens to an oriented surface ΣΓwith boundary.
To build ΣΓ, replace each edge of G by a strip and glue these strips together
at the vertices according to the fat structure.
Proposition 1. The boundary components of ΣΓcorrespond to the cycles
of ω = σ · i which is a permutation on the set of half-edges.
Proof. Any half-edge A corresponds to one side of the strip of the edge
{A,i(A)} and hence corresponds to part of a boundary component ∂iΣΓ.
Following this boundary along A leads to its target vertex v = s(i(A)) where
it follows the next strip corresponding to σ(i(A)) = ω(A) and so on.
?
The orbits of the permutation ω are called the boundary cycles of a
fat graph. Since σ = ω · i, the permutation ω completely determines the
fat structure.
Example 2. Changing the fat structure σ affects the induced surface in a
fundamental way. In figure 1, the fat graphs Γ and?Γ have the same under-
are respectively,
lying graph but they have different fat structures. Their boundary cycles
ωΓ= (ABCABC)
ωeΓ= (AC)(BA)(CB).
Hence the surface ΣeΓhas three boundary components and genus 0 while the
original ΣΓhas a single boundary component and genus 1.
A morphism of fat graph
ϕ : (G,ω) ?−→ (?G, ? ω)
is a morphism of combinatorial graphs such that ϕ(ω) = ? ω. Hence ? ω is
or by skipping A if ϕ(A) is a vertex. In [15], Igusa built a category Fat whose
objects are fat graphs with no univalent or bivalent vertices and with
ordered boundary cycles. The morphism of Fat are morphisms of fat graph
obtained from ω by replacing an half-edge A with ϕ(A) if it is an half-edge
Page 36
36V´ERONIQUE GODIN
Proof. The space |Fatb
resenting the movement of one of the leaf around its boundary cycle. We
define a parameterized version of the pair of pants gluing µ as follows.
ξ : |Fatb
0,3| is homotopy equivalent to (S1)3each circle rep-
0,3| × |Fatb
g,1| × |Fatb
h,1|
−→
|Fatb
?
g+h,1|
((t0,t1,t2),Γb,?Γb)
?−→
Γb
#
t0,t1,t2
?Γb?
.
The new metric bordered fat graph, which illustrated in figure 9, is obtained
by removing the leaf-edges of Γband?Γband by adding a new edge K of length
t2-th of the way around the boundary cycle of?Γb. Finally a leaf is attached
of the edge K.
As in figure 9, we construct a Z/2-equivariant map,
one which is attached t1-th of the way around the boundary cycle of Γband
t0-th of the way around the new boundary whose starting point is middle
f : S1× X × X −→ C(2) × X × X −→ |Fatb
X × X, the two disks of C(2) and two of the boundary components of S0,3.
Let e be the one-simplex corresponding to the top half of S1. For any k-cycle
u of X, the chain e⊗u⊗u maps to a cycle of C2k+1(X;R). The Kudo-Araki
is defined to be the homology class represented by this cycle.
The map
0,3| × X × X
ξ
−→ X.
The group Z/2 acts on these spaces : it exchanges the two coordinates of
C1(S1) ⊗ Ck(X) −→ C1((S1)3) ⊗ Ck(X)⊗2
(π ⊗ e ⊗ π ⊗ u ⊗ u) + ((−e) ⊗ 0 ⊗ π ⊗ u ⊗ u) + (0 ⊗ 0 ⊗ e ⊗ u ⊗ u)
∼=
−→ C1(S1)⊗3⊗ Ck(X)⊗2
sends e ⊗ u to
Let γ ∈ C1(S1) be the entire circle. Using that f is Z/2-equivariant,
Q1(u) = ξ(0 × 0 × γ × u × u) + ξ(−e × 0 × π × u × u).
Using proposition 25 and lemma 13, this gives the result.
?
Example 6. The Dyer-Lashoff operation
Q1: H∗(M(S1,1;∂)) −→ H∗(M(S2,1;∂))
0]) = µ∗([Γb
maps the generator [Γb
0] for H0(M(S1,1;∂);Z) to
q1([Γb
0],ζ([Γb
0])) +
?
(Φi([Γb
i
Φi([Γb
0#Γb
0])
= −6 ∗ [Γb
0#Γb
1] +
?
i
0#Γb
0])) = −12 + 5 = 3.
To see that the second term gives 5, notice that it is the lifting of Aut(Γ2)
of (4.3) to BGC∗. The result then follows from the proof of proposition 22.
Similarly q1maps H1(M(S1,1;∂);Z/2) surjectively to the cokernel of the
map
H3(M(S2,1;∂);Z) ⊗ Z/2 ֒−→ H3(M(S2,1;∂);Z/2).
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THE UNSTABLE INTEGRAL HOMOLOGY OF THE MAPPING CLASS GROUPS 37
We define
ψ1 : BGCp⊗ BGCq−→ BGCp+q+1
µ?[Γb,o],[?Γb, ˜ o]??
Proposition 28. The map ψ1induces the Browder operation on the ho-
mology of the bordered mapping class groups.
by sending [Γb,o] ⊗ [?Γb, ˜ o] to
−ξ
?
+ µ
?
ζ([Γb,o]),[?Γb, ˜ o]
?
+ (−1)pµ
?
[Γb,o],ζ?[?Γb, ˜ o]??
.
Proof. Using the notation of the proof of proposition 27, the Browder oper-
ation is defined by
ψ1(u ⊗ v) = f(γ ⊗ u ⊗ v)
where again γ ∈ C1(S1) is the entire circle. Since γ is first mapped to
(γ−1× γ × (γ + π))
≃
(γ−1× 0 × π) + (0 × γ × π) + (0 × 0 × γ + π)
(γ−1× 0 × 0) + (0 × γ × 0) + (0 × 0 × γ)
≃
in (S1)3, it suffices to find the image under ξ∗of the chains
(γ−1×0×0×[Γb]×[?Γb])
Example 7. The Browder operation Ψ1gives
(0×γ×0×[Γb]×[?Γb])(0×0×γ×[Γb]×[?Γb])
which is done as in the previous proof.
?
H0(M(S1,1;∂)) ⊗ H0(M(S1,1;∂))
6
−→
0
−→
H1(M(S2,1;∂))
H0(M(S1,1;∂)) ⊗ H1(M(S1,1;∂))
0
−→
H2(M(S2,1;∂))
H1(M(S1,1;∂)) ⊗ H1(M(S1,1;∂))
H3(M(S2,1;∂)).
In particular, this shows that 6 ∈ H1(M(S2,1;∂)) is unstable.
5.3. Higher Kudo-Araki-Dyer-Lashof operations. Let Sg,1be a sur-
face of genus g with a single boundary component. As before, let M∞be
M(S1,1;∂)
−→ M(S2,1;∂)
mapping class groups gives rise to Dyer-Lashoff-Araki-Kudo operations
the direct limit of the following sequence of groups.
Φ1
Φ2
−→ M(S3,1;∂) −→ ...
The infinite loop structure of Tillmann [28] on the homology of the stable
?Qi,p: Hn(M∞,δ;Z/p) −→ Hpn+i(M∞,δ;Z/p).
Tillmann constructed homological operation in [5] using simple geometric
construction and Harer stability.These operations are the Kudo-Araki-
Dyer-Lashof operations associated to the infinite loop structures on M∞.
these operations. These will give homomorphisms
Before the existence of such an infinite loop structure was known, Cohen and
In this section, we construct an unstable and combinatorial version of
Qi,p: Hn(M(Sg,1;∂);Z/p) −→ Hpn+i(M(Spg,1);Z/p).
Page 38
38V´ERONIQUE GODIN
Figure 10. The embedding Z/5 →?
The infinite loop space operations are obtained from these by composing
with the stable splitting
M1
0,5
s : Hn(M(S)) −→ Hn(M(S;∂))
which exists from the proof of corollary 16.
Fix a prime p and a surface S0,p+1of genus 0 with p + 1 boundary com-
ponents. Fix parameterizations {φi} of the first p boundary components.
Consider the group
?
M1
0,p= π0Diff+(S0,p+1;{φi})
of isotopy classes of diffeomorphisms of S which preserve the last boundary
component as a set. These diffeomorphisms are allowed to permute the first
p boundary components but they must preserve the parameterizations. Us-
ing the action of the elements of?
?
The rotation illustrated in figure 10 gives an inclusion Z/p →?
we get
?
which gives a chain map
?C∗(BM(Sg,1;∂))?⊗p Ψ∗
For a cycle α ∈ Ck(M(Sg,1;∂);Z/p), we define
Qi([α]) = θ?ei⊗ x⊗p?∈ Hpk+i(M(Spg,1;∂);Z/p)
where each eiis one of the generator of the standard resolution of Z/p over
itself.
Consider the following category C(p,g) which is the wreath product of
the category Fatbwith one-object category C(p) whose morphisms are Z/p.
M1
0,pon the first p boundary components,
we get a wreath product
M1
0,p
?
M(Sg,1;∂) =?
M1
0,p⋉?M(Sg,1;∂)p?.
M1
0,p. By
gluing the p surfaces Sg,1onto the first p boundary components of S0,p+1,
(5.2)
Z/p
M(Sg,1;∂) −→?
M1
0,p
?
M(Sg,1;∂) −→ M(Spg,1)
θ : C∗(EZ/p) ⊗
Z/p
−→ C∗(M(Spg,1))
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THE UNSTABLE INTEGRAL HOMOLOGY OF THE MAPPING CLASS GROUPS 39
The objects of C(p,g) are p-tuples of bordered fat graphs. The morphisms
of C(p,g) are tuples
(i,ϕ1...ϕp) : (Γb
1,...Γb
p) −→ (?Γb
1,...?Γb
p)
Here ϕj: Γb
j→?Γb
j+iis a morphism of bordered fat graphs. By construction
?
Following the contruction of the pair of pants functor µ, we define
|C(p,g)| ≃ B
Z/p
?
M(S;∂)
?
µp: C(p,g) −→ Fat1
pg.
A p-tuple (Γb
identifying the leaf-vertices l1,...,lp. A morphism of C(p,g) induces a mor-
phism of Fat in the obvious way. The functor µprealizes the homomorphism
of groups of (5.2). We therefore get the homomorphism θ by first using the
shuffle map
1...Γb
p) is sent to the unbordered fat graph Γnew obtained by
C∗(E(p)) ⊗
Z/p
?C∗(Fatb
g,1)?⊗p−→ C∗
?
E(p) ⊗
Z/p
?Fatb
g,1
??
= C∗(C(p,g))
−→ C∗(Fatpg,1)
Here E(p) is the universal cover of C(p). It has p object and a single mor-
phism between any two of them. The chains ei∈ Ck(E(p);Z/p)) which are
the image of the generators of the standard resolution are as follows.
e2k
=
?
(1,i1,1,i2,...,ik,1) = (0
1≤i1...ik≤p−1
?
(i1,1,i2,1...,ik,1) = (0
i1
→ i1
i1
→ (i1+ 1)
1
→ (i1+ 1)
1
→ (i1+ 2)...)
i2
→ ...)
e2k+1
=
i1...ik
1
→ 1
Example 8. Consider the operations
Qi,2: H0(M(S1,1;∂);Z/2) −→ Hi(M(S2,1);Z/2).
page 30) and let ϕ be the generator of Aut(?Γ2)∼= Z/2. By definition
Qi,2([Γb
Let [Γb
0] be the generator of H0(M(S1,1;∂), let?Γ2be again the fat graph of
0]) = [(ϕ,ϕ,... ,ϕ)] ∈ Hi(Fat;Z/2)
In proposition 22, we have shown that for i odd
[(ϕ,ϕ,... ,ϕ)] = 5 ∈ Z/10 ⊂ H∗(M(S2,1;Z)
Page 40
40V´ERONIQUE GODIN
By the proof of proposition 18, we know that the E∞term of the appropriate
Lerray-Serre spectral sequence with Z/2-coefficients is
Z/20
Z/2 ⊕ Z/20
Z/2 ⊕ Z/2 ⊕ Z/20
Z/2 ⊕ Z/2 ⊕ Z/2
Z/2 ⊕ Z/20
Z/20
Z/20
Z/2
···
...
...
...
...
...
Z/2 ⊕ Z/2
Z/2 ⊕ Z/2
Z/2
Z/2
←−
Using the argument of section 4.3, we can show that
Q2k,2([Γb
0]) ∈ H2k(M(S2,2;∂);Z/2)
generates the Z/2’s in the marked row.
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Department of Mathematics, Harvard University, Cambridge, MA 02138
E-mail address: godin@math.harvard.edu