arXiv:math/0506400v2 [math.GT] 10 Dec 2008
An infinite genus mapping class group and stable cohomology∗
Louis Funar Christophe Kapoudjian
Institut Fourier BP 74, UMR 5582Laboratoire Emile Picard, UMR 5580
University of Grenoble IUniversity of Toulouse III
38402 Saint-Martin-d’H` eres cedex, France 31062 Toulouse cedex 4, France
December 11, 2008
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.
1.1 Statements of the main results
The tower of all extended mapping class groups was considered first by Moore and Seiberg () 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 () 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 , we introduced the universal mapping class group in genus zero B. The latter is an extension of the
Thompson’s group V (see ) by the infinite spherical pure mapping class group. We proved in  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
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., ). 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 , 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 ), 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
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
1.2.1 The 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.
Proof. Denote by Z the set of symmetric Hilbert-Schmidt operators H− → H+ with norm < 1. Clearly,
Z is a contractible subspace of the Banach space of Hilbert-Schmidt operators. The group Spres(Hr) acts
transitively and continuously (see  p. 177) on Z by means of
g(S) = (Φ(g)S + Ψ(g))(Ψ(g)S + Φ(g))−1∈ Z, for g ∈ Spres(Hr),S ∈ Z
isomorphic to U(H+). By a result of Kuiper (), U(H+) is contractible. The claim is now a consequence
of the contractibility of Spres(Hr)/U(H+)∼= Z.
Proposition 5.2. For each integer n ∈ Z, there is a well-defined continuous cocycle Cndefined on Spres(Hr),
with values in C∗, such that
Cn(g,g′) = det(Φ(g)Φ(g′)Φ(gg′)−1)
The stabilizer of S = 0 is the group of matrices
such that Φ is unitary in H+. Thus, it is
|Cn|may be lifted to a real cocycle ? ςn: Spres(Hr) × Spres(Hr) −→ R such that
The restriction ς1of ? ς1to Sp(2∞,R) defines an integral cohomology class [ς1] ∈ H2(Sp(2∞,R),Z).
Proof. In fact, Φ(g)−1Φ(g′)−1Φ(gg′) = 1 + (Φ(g′)−1Φ(g)−1Ψ(g)(Ψ(g′)). But, according to (, p. 168) we
||Φ−1(g)Ψ(g)|| < 1 and ||Ψ(g′)Φ(g′)−1|| < 1
Thus, there is a non-ambiguous definition of (Φ−1(g)Φ(gg′)Φ−1(g′))
ςn(g,g′), for all g,g′∈ Spres(Hr)
n given by an absolutely convergent
The existence of ςnis now an immediate consequence of the preceding lemma.
The map ℓ : g ∈ Sp(2∞,R) ?→ ℓ(g) =
|det(Φ(g))|is well-defined, so that the cocycle
(g,g′) ∈ Sp(2∞,R) × Sp(2∞,R) ?→ e2iπς1(g,g′)
is the coboundary of ℓ. This proves that the cohomology class of ς1restricted to Sp(2∞,R) is integral.
Remark 5.1.1. The restrictions of the real cocycles ςnon the finite dimensional Lie group Sp(2g,R) are
those constructed by Dupont-Guichardet-Wigner (see ). In fact, the authors of  proved that the
cohomology class of the restriction of ς1to Sp(2g,R) is integral, and is the image in H2(Sp(2g,R),R) of
the generator of H2
bor(Sp(2g,R),Z) = Z, the second group of borelian cohomology of Sp(2g,R). They
prove also that it is the image of the first Chern class c1(BU(g,C)) by the composition of maps
H2(BU(g,C),Z) ≈ H2(BSp(2g,R),Z) → H2(BSp(2g,R)δ,Z) ≈ H2(Sp(2g,R),Z) → H2(Sp(2g,R),R),
where BSp(2g,R)δis the classifying space of Sp(2g,R) as a discrete group.
2. The remark above implies that the map
H∗(BU,Z) ≈ H∗(BSp(2∞,R),Z) → H∗(Sp(2∞,R),Z).
sends the first universal Chern class c1(BU) onto [ς1] ∈ H2(Sp(2∞,R),Z). Further, the symplectic
representation ρ : PM → Sp(2∞,R) maps [ς1] onto the generator c1(PM) of H2(PM,Z).
3. According to (, Theorem 6.2.3), the Berezin-Segal-Shale-Weil cocycle is the complex conjugate of
the cocycle C−1
Theorem 5.1. Let [ˆ ς1] ∈ H2(Spres(Hr),R) be the cohomology class of ˆ ς1. The pull-back of [ˆ ς1] in H2(M,R)
by the representation ˆ ρ of Theorem 1.3 is integral, and is the natural image of the generator c1(M) of
H2(M,Z) in H2(M,R).
Proof. Let ι : PM → M and j : Sp(2∞,R) → Spres(Hr) be the natural embeddings. Plainly, j ◦ ρ = ˆ ρ ◦ ι.
Since j∗: H2(Spres(Hr),R) → H2(Sp(2∞,R),R) maps [ˆ ς1] onto [ς1], one has ι∗(ˆ ρ∗[ˆ ς1]) = ρ∗[ς1].
us denote by ¯ c1(PM) (respectively ¯ c1(M)) the image of c1(PM) (respectively c1(M)) in H2(PM,R)
(respectively H2(M,R)). According to Remark 5.1, 2., ρ∗[ς1] = ¯ c1(PM). By Proposition 3.1, ι∗(¯ c1(PM)) =
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