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

by Rasmus Villemoes
PhD Dissertations October 2009

Let Σ be a compact surface, let Γ denote its mapping class group, and let G be a Lie group. Then Γ acts on the space

MG=Hom(π1Σ,G)/G

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

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

In the thesis it is proved that H1(Γ,V)=0 for each of the above-mentioned representations. The proofs of these theorems roughly follows the same recipe: (a) Find a 'basis' B for the vector space V represented by geometric objects on the surface, such that the Γ-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 VMap(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) is injective, which is the same as proving that

H0(Γ,V)H0(Γ,V/V)(1)

is surjective.

It is known that one may use the set of 'multicurves' on Σ in case (1a), whereas the integral homology of Σ, 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 Γ-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 v is an 'almost invariant' element of V (in the sense that vγvV for every γΓ), then v is actually almost equal to an invariant element of V (in the sense that there exists wV such that vwH0(Γ,V)).

Format available: PDF (876 KB)
Thesis advisor: Jørgen Ellegaard Andersen