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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.

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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.