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arXiv:math/0506400v2 [math.GT] 10 Dec 2008

An infinite genus mapping class group and stable cohomology∗

Louis FunarChristophe Kapoudjian

Institut Fourier BP 74, UMR 5582 Laboratoire Emile Picard, UMR 5580

University of Grenoble IUniversity of Toulouse III

38402 Saint-Martin-d’H` eres cedex, France31062 Toulouse cedex 4, France

e-mail: funar@fourier.ujf-grenoble.fr

e-mail: ckapoudj@picard.ups-tlse.fr

December 11, 2008

Abstract

We exhibit a finitely generated group M whose rational homology is isomorphic to the rational stable

homology of the mapping class group. It is defined as a mapping class group associated to a surface

S∞ of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g

with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of M into the

restricted symplectic group Spres(Hr) of the real Hilbert space generated by the homology classes of

non-separating circles on S∞, which generalizes the classical symplectic representation of the mapping

class groups. Moreover, we show that the first universal Chern class in H2(M,Z) is the pull-back of the

Pressley-Segal class on the restricted linear group GLres(H) via the inclusion Spres(Hr) ⊂ GLres(H).

2000 MSC Classification: 57 N 05, 20 F 38, 22 E 65,81 R 10.

Keywords: mapping class groups, infinite surface, Thompson group, stable cohomology, Chern class,

restricted symplectic group.

1Introduction

1.1Statements of the main results

The tower of all extended mapping class groups was considered first by Moore and Seiberg ([19]) as part

of the conformal field theory data. This object is actually a groupoid, which has been proved to be finitely

presented (see [1, 2, 7, 15]). When seeking for a group analog Penner ([24]) investigated a universal map-

ping class group which arises by means of a completion process and which is closely related to the group of

homeomorphisms of the circle, but it seems to be infinitely generated.

In [8], we introduced the universal mapping class group in genus zero B. The latter is an extension of the

Thompson’s group V (see [5]) by the infinite spherical pure mapping class group. We proved in [8] that the

group B is finitely presented and we exhibited an explicit presentation. Our main difference with the previous

attempts is that we consider groups acting on infinite surfaces with a prescribed behaviour at infinity that

comes from actions on trees.

Following the same kind of approach, we propose a treatment of the arbitrary genus case by introducing a

mapping class group M, called the asymptotic infinite genus mapping class group, that contains a large part

of the mapping class groups of compact surfaces with boundary. More precisely, the group M contains all

the pure mapping class groups PM(Σg,n) of compact surfaces Σg,nof genus g with n boundary components,

for any g ≥ 0 and n > 0. Its construction is roughly as follows. Let S denote the surface obtained by taking

the boundary of the 3-dimensional thickening of the complete trivalent tree, and further let S∞be the result

of attaching a handle to each cylinder in S that corresponds to an edge of the tree (see figure 1). Then M

is the group of mapping classes of those homeomorphisms of S∞ which preserve a certain rigid structure

at infinity (see Definition 1.3 for the precise definition). This rigidity condition essentially implies that M

∗This version: January 30, 2008.L.F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.

electronically at http://www-fourier.ujf-grenoble.fr/∼funar

Available

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induces a group of transformations on the set of ends of the tree, which is isomorphic to Thompson’s group

V . The relation between both groups is enlightened by a short exact sequence 1 → PM → M → V → 1,

where PM is the mapping class group of compactly supported homeomorphisms of S∞. The latter is an

infinitely generated group. Our first result is:

Theorem 1.1. The group M is finitely generated.

The interest in considering the group M, outside the framework of the topological quantum field theory

where it can replace the duality groupoid, is the following homological property:

Theorem 1.2. The rational homology of M is isomorphic to the stable rational homology of the (pure)

mapping class groups.

As a corollary of the argument of the proof (see Proposition 3.1), the group M is perfect, and H2(M,Z) = Z.

For a reason that will become clear in what follows, the generator of H2(M,Z)∼= Z is called the first uni-

versal Chern class of M, and is denoted c1(M).

Let Mgbe the mapping class group of a closed surface Σgof genus g. We show that the standard represen-

tation ρg: Mg→ Sp(2g,Z) in the symplectic group, deduced from the action of Mgon H1(Σg,Z), extends

to the infinite genus case, by replacing the finite dimensional setting by concepts of Hilbertian analysis. In

particular, a key role is played by Shale’s restricted symplectic group Spres(Hr) on the real Hilbert space Hr

generated by the homology classes of non-separating closed curves of S∞. We have then:

Theorem 1.3. The action of M on H1(S∞,Z) induces a representation ρ : M → Spres(Hr).

The generator c1 of H2(Mg,Z) is called the first Chern class, since it may be obtained as follows (see,

e.g., [20]). The group Sp(2g,Z) is contained in the symplectic group Sp(2g,R), whose maximal compact

subgroup is the unitary group U(g). Thus, the first Chern class may be viewed in H2(BSp(2g,R),Z). It

can be first pulled-back on H2(BSp(2g,R)δ,Z) = H2(Sp(2g,R),Z) and then on H2(Mg,Z) via ρg. This is

the generator of H2(Mg,Z). Here BSp(2g,R)δdenotes the classifying space of the group Sp(2g,R) endowed

with the discrete topology.

The restricted symplectic group Spres(Hr) has a well-known 2-cocycle, which measures the projectivity of

the Berezin-Segal-Shale-Weil metaplectic representation in the bosonic Fock space (see [22], Chapter 6 and

Notes p. 171). Contrary to the finite dimension case, this cocycle is not directly related to the topology of

Spres(Hr), since the latter is a contractible Banach-Lie group. However, Spres(Hr) embeds into the restricted

linear group of Pressley-Segal GL0

res(H) (see [25]), where H is the complexification of Hr, which possesses

a cohomology class of degree 2: the Pressley-Segal class PS ∈ H2(GL0

homotopic model of the classifying space BU, where U = lim

res(H),C∗). The group GL0

n→∞U(n,C), and the class PS does correspond to

res(H) is a

the universal first Chern class. Its restriction on Spres(Hr) is closely related to the Berezin-Segal-Shale-Weil

cocycle, and reveals the topological origin of the latter. Via the composition of morphisms

M −→ Spres(Hr) ֒→ GL0

res(H),

we then derive from PS an integral cohomology class on M (see Theorem 5.1 for a more precise statement):

Theorem 1.4. The Pressley-Segal class PS ∈ H2(GL0

c1(M) ∈ H2(M,Z).

Acknowledgements. The authors are indebted to Vlad Sergiescu for enlighting discussions and particularly

for suggesting the existence of a connection between the first universal Chern class of M and the Pressley-

Segal class. They are thankful to the referees for suggestions improving the exposition.

res(H),C∗) induces the first universal Chern class

1.2Definitions

1.2.1The infinite genus mapping class group M

Set M(Σg,n) for the extended mapping class group of the n-holed orientable surface Σg,nof genus g, consisting

of the isotopy classes of orientation-preservinghomeomorphisms of Σg,nwhich respect a fixed parametrization

of the boundary circles, allowing them to be permuted among themselves.

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circles of pants decompositions

graph drawn on the surface

Figure 1: The infinite genus surface S∞with its canonical rigid structure

We wish to construct a mapping class group, containing all mapping class groups M(Σg,n).

impossible to construct such a group, but if one relaxes slightly our requirements then we could follow our

previous method used for the genus zero case in [8].

It seems

The choice of the extra structure involved in the definitions below is important because the final result

might depend on it. For instance, using the same planar punctured surface but different decompositions one

obtained in [9] two non-isomorphic braided Ptolemy-Thompson groups.

Definition 1.1 (The infinite genus surface S∞). Let T be the complete trivalent planar tree and S be the

surface obtained by taking the boundary of the 3-dimensional thickening of T .

By grafting an edge-loop (i.e. the graph obtained by attaching a loop to a boundary vertex of an edge) at the

midpoint of each edge of T , one obtains the graph T∞. The surface S∞is the boundary of the 3-dimensional

thickening of T∞.

The graph T (respectively T∞) is embedded in S (respectively S∞) as a cross-section of the fiber projection,

as indicated on figure 1. Thus, S∞is obtained by removing small disks from S centered at midpoints of edges

of T and gluing back one holed tori Σ1,1, called wrists which correspond to the thickening of edge-loops.

It is convenient to assume that T is embedded in a horizontal plane, while the edge-loops are in vertical

planes (see figure 1).

Definition 1.2 (Pants decomposition of S∞). A pants decomposition of the surface S∞ is a maximal

collection of distinct nontrivial simple closed curves on S∞ which are pairwise disjoint and non-isotopic.

The complementary regions (which are 3-holed spheres) are called pairs of pants.

By construction, S∞is naturally equipped with a pants decomposition, which will be referred to below as the

canonical (pants) decomposition, as shown in figure 1:

• the wrists are decomposed using a meridian circle and the boundary circle of Σ1,1.

• there is one pair of pants for each edge, which has one boundary circle for attaching the wrists, and

two circles to grip to the other type of pants. We call them edge pants.

• there is one pair of pants for each vertex of the tree, called vertex pants.

A pants decomposition is asymptotically trivial if outside a compact subsurface of S∞, it coincides with the

canonical pants decomposition.

Definition 1.3.

vertex type pair of pants from the canonical decomposition and moreover, if one boundary circle from

a vertex type pants is contained in Σ then the entire pants is contained in Σ. In particular, S∞− Σ

has no compact components.

1. A connected subsurface Σ of S∞ is admissible if all its boundary circles are from

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2. Let ϕ be a homeomorphism of S∞. One says that ϕ is asymptotically rigid if the following conditions

are fulfilled:

• There exists an admissible subsurface Σg,n⊂ S∞such that ϕ(Σg,n) is also admissible.

• The complement S∞−Σg,nis a union of n infinite surfaces. Then the restriction ϕ : S∞−Σg,n→

S∞−ϕ(Σg,n) is rigid, meaning that it maps the pants decomposition into the pants decomposition

and maps T∞∩ (S∞− Σg,n) onto T∞∩ (S∞− ϕ(Σg,n)). Such a surface Σg,nis called a support

for ϕ.

One denotes by M = M(S∞) the group of asymptotically rigid homeomorphisms of S∞up to isotopy

and call it the asymptotic mapping class group of infinite genus.

In the same way one defined the asymptotic mapping class group M(S), denoted by B in [8].

Remark 1.1. In genus zero (i.e. for the surface S) a homeomorphism between two complements of admissible

subsurfaces which maps the restrictions of the tree T one into the other is rigid, thus preserves the isotopy

class of the pants decomposition. This is not anymore true in higher genus: the Dehn twist along a longitude

preserves the edge-loop graph but it is not rigid, as a homeomorphism of the holed torus.

Remark 1.2. Notice that, in general, rigid homeomorphisms ϕ do not have an invariant support i.e. an

admissible Σg,nsuch that ϕ(Σg,n) = Σg,n. Take for instance a homeomorphism which translates the wrists

along a geodesic ray in T .

Remark 1.3. Any admissible subsurface Σg,n ⊂ S∞ has n = g + 3. Moreover S∞ is the ascending union

∪∞

The admissible subsurfaces will be Σkg,k+3. The asymptotic mapping class group obtained this way is finitely

generated by small changes in the proof below.

g=1Σg,g+3. Instead of the wrist Σ1,1use a surface of higher genus Σg,1and the same definitions as above.

Remark 1.4. The surface S∞ contains infinitely many compact surfaces of type (g,n) with at least one

boundary component. For any such compact subsurface Σg,n⊂ S∞, there is an obvious injective morphism

i∗: PM(Σg,n) ֒→ PM ⊂ M. However, the morphism i∗: M(Σg,n) ֒→ M is not always defined. Indeed,

it exists if and only if the n connected components of S∞\ Σg,n are homeomorphic to each other, by

asymptotically rigid homeomorphisms.

In particular, for any admissible subsurface Σg,n (hence n = g + 3), i∗ extends to an injective morphism

i∗: M(Σg,n) ֒→ M defined by rigid extension of homeomorphisms of Σg,nto S∞.

1.2.2The group M and the Thompson groups

Definition 1.4.

1. Let T be the planar trivalent tree. A partial tree automorphism of T is an isomor-

phism of graphs ϕ : T \τ1→ T \τ2, where τ1and τ2are two finite trivalent subtrees of T (each vertex

except the leaves are 3-valent). A connected component of T \τ1or T \τ2is a branch, that is, a rooted

planar binary tree whose vertices are 3-valent, except the root, which is 2-valent. Each vertex of a

branch has two descendant edges, and given an orientation to the plane, one may distinguish between

the left and the right descendant edges. A partial automorphism ϕ : T \τ1→ T \τ2is planar if it maps

each branch of T \ τ1onto the corresponding branch of T \ τ2by respecting the left and right ordering

of the edges.

2. Two planar partial automorphisms ϕ : T \ τ1→ T \ τ2and ϕ′: T \ τ′

is denoted ϕ ∼ ϕ′, if and only if there exists a third ϕ′′: T \ τ′′

τ2∪ τ′

3. If ϕ and ϕ′are planar partial automorphisms, one can find ϕ0∼ ϕ and ϕ′

of ϕ0and the target of ϕ′

The set of equivalence classes of such automorphisms endowed with the above internal law, is a group

with neutral element the class of idT. This is the Thompson group V .

Remark 1.5. We warn the reader that our definition of the group V is different from the standard one (as

given in [5]). Nevertheless, the present group V is isomorphic to the group denoted by the same letter in [5].

1→ T \ τ′

1→ T \ τ′′

2are equivalent, which

2such that τ1∪ τ′

1⊂ τ′′

1,

2⊂ τ′′

2and ϕ|T \τ′′

1= ϕ′

|T \τ′′

1= ϕ′′.

0∼ ϕ′such that the source

0] is well defined, as is easy to check.

0coincide. The product [ϕ]·[ϕ′] = [ϕ0◦ϕ′

We introduce Thompson’s group T, the subgroup of V acting on the circle (see [11]), which will play a key

role in the proofs.

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Definition 1.5 (Ptolemy-Thompson’s group T). Choose a vertex v0of T . Each g ∈ V may be represented

by a planar partial automorphism ϕ : T \τ1→ T \τ2such that v0belongs to τ1∩τ2. Let D1(respectively D2)

be a disk containing τ1(respectively τ2), whose boundary circle S1(respectively S2) passes through the leaves

of τ1(respectively τ2), giving to them a cycling ordering. If ϕ preserves this cycling ordering, which amounts

to saying that the bijection from the set of leaves of τ1 onto the set of leaves of τ2 can be extended to an

orientation preserving homeomorphism from S1onto S2, then any other ϕ′equivalent to ϕ also does, and one

says that g itself is circular. The subset of circular elements of V is a subgroup, called the Ptolemy-Thompson

group T.

Proposition 1.1. Set PM for the inductive limit of the pure mapping class groups of admissible subsurfaces

of S∞. We have then the following exact sequences:

1 → PM → M → V → 1.

Proof. Let ϕ be an asymptotically rigid homeomorphism of S∞ and Σg,n a support for ϕ. Then it maps

T∞∩(S∞−Σg,n) onto T∞∩(S∞−ϕ(Σg,n)), hence T ∩(S∞−Σg,n) onto T ∩(S∞−ϕ(Σg,n)) by forgetting

the action on the edge-loops. This may be identified with a planar partial automorphism φ : T \τ1→ T \τ2.

The map [ϕ] ∈ M → [φ] ∈ V is a group epimorphism. The kernel is the subgroup of isotopy classes of

homeomorphisms inducing the identity outside a support, and hence is the direct limit of the pure mapping

class groups.

Remark 1.6. In [8] we prove the existence of a similar short exact sequence relating B to V , which splits

over the Ptolemy-Thompson group T. It is worth noticing that the present extension of V is not split over

T.

2 The proof of theorem 1.1

2.1Specific elements of M

Recall that S∞has a canonical pants decomposition, as shown in figure 1. We fix an admissible subsurface

A = Σ1,4which contains a central wrist and an admissible B = Σ0,3⊂ Σ1,4which is not adjacent to the

wrist.

Let us consider now the elements of M described in the pictures below. Specifically:

• Let γ be a circle contained inside B and parallel to the boundary curve labeled 3. Let t be the right

Dehn twist around γ. This means that, given an outward orientation to the surface, t maps an arc

crossing γ transversely to an arc which turns right as it approaches γ. The dashed arcs (also called

seams) on the left hand side picture figure out the boundary of the visible side of B. Their images by

t are represented on the right hand side picture,

t

3

12

21

3

• π is the braiding, acting as a braid in M(Σ0,3), with the support B. It rotates the circles 1 and 2 in

the horizontal plane (spanned by the circles) counterclockwise.

Assume that B is identified with the complex domain {|z| ≤ 7,|z −3| ≥ 1,|z +3| ≥ 1} ⊂ C. A specific

homeomorphism in the mapping class of π is the composition of the counterclockwise rotation of 180

degrees around the origin — which exchanges the small boundary circles labeled 1 and 2 in the figure

— with a map which rotates of 180 degrees in the clockwise direction each boundary circle. The latter

can be constructed as follows.

Let A be an annulus in the plane, which we suppose for simplicity to be A = {1 ≤ |z| ≤ 2}. The

homeomorphism DA,C acts as the counterclockwise rotation of 180 degrees on the boundary circle C

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