# PhD defence by Rasmus Villemoes (2009)

Title: Cohomology of Mapping Class Groups with Coefficients in Functions on Moduli Spaces

2009.10.08 | Christine Dilling

 Date Thu 08 Oct Time 14:15 — 16:15 Location

Abstract:

Let ΣΣ be a compact surface, let ΓΓ denote its mapping class group, and let GG be a Lie group. Then ΓΓ acts on the space

MG=Hom(π1Σ,G)/GMG=Hom(π1Σ,G)/G

of GG-representations of the fundamental group of ΣΣ, also known as the moduli space of flat GG-connections over ΣΣ. This action induces representations of ΓΓ on various 'large' vector spaces:

(1) When G=SL2(C)G=SL2(C), MGMG is an affine algebraic set. Since the action of ΓΓ is algebraic, there is an induced action on the space O(MG)O(MG) of regular functions on MGMG.
(2) When GG is the circle group U(1)U(1), MGMG is a smooth, compact, symplectic manifold, and ΓΓ acts by symplectomorphisms. Thus both C∞(MG)⊆L2(MG)C∞(MG)⊆L2(MG) are unitary representations of ΓΓ.

In the thesis it is proved that H1(Γ,V)=0H1(Γ,V)=0 for each of the above-mentioned representations. The proofs of these theorems roughly follows the same recipe: (a) Find a 'basis' BB for the vector space VV represented by geometric objects on the surface, such that the ΓΓ-action is given by permuting this basis; (b) write down the action of a Dehn twist on a basis element; (c) prove that the inclusion V→Map(B,C)=V∗V→Map(B,C)=V∗ induces the zero map on cohomology; and finally (d) use well-known relations in the mapping class group to deduce that the map H1(Γ,V)→H1(Γ,V∗)H1(Γ,V)→H1(Γ,V∗) is injective, which is the same as proving that

H0(Γ,V∗)→H0(Γ,V∗/V)(1)H0(Γ,V∗)→H0(Γ,V∗/V)(1)

is surjective.

It is known that one may use the set of 'multicurves' on ΣΣ in case (1a), whereas the integral homology of ΣΣ, in the guise of 'pure phase functions', can be used in (1b). In both cases, the action of a Dehn twist has a well-known and simple description. Step (c) can, via Shapiro's Lemma, be translated into a question about the ΓΓ-stabilizer of basis elements, and that step is also relatively easy. Step (d) is the most technical. Proving that (1) is surjective amounts to proving that if vv is an 'almost invariant' element of V∗V∗ (in the sense that v−γv∈Vv−γv∈V for every γ∈Γγ∈Γ), then vv is actually almost equal to an invariant element of V∗V∗ (in the sense that there exists w∈Vw∈V such that v−w∈H0(Γ,V∗)v−w∈H0(Γ,V∗)).