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

By Rasmus Villemoes
PhD Dissertations
October 2009
Abstract:

Let $\Sigma$ be a compact surface, let $\Gamma$ denote its mapping class group, and let $G$ be a Lie group. Then $\Gamma$ acts on the space

$M_G = \mathrm{Hom}(\pi_1 \Sigma, G)/G$

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

(1)
When $G = \mathrm{SL}_2(\mathbb{C})$, $M_G$ is an affine algebraic set. Since the action of $\Gamma$ is algebraic, there is an induced action on the space $\mathcal{O}(M_G)$ of regular functions on $M_G$.
(2)
When $G$ is the circle group $\mathrm{U}(1)$, $M_G$ is a smooth, compact, symplectic manifold, and $\Gamma$ acts by symplectomorphisms. Thus both $C^\infty(M_G) \subseteq L^2(M_G)$ are unitary representations of $\Gamma$.

In the thesis it is proved that $H^1(\Gamma, V) = 0$ 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 $\Gamma$-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 $V\to \mathrm{Map}(B, \mathbb{C}) = V^*$ induces the zero map on cohomology; and finally (d) use well-known relations in the mapping class group to deduce that the map $H^1(\Gamma, V) \to H^1(\Gamma, V^*)$ is injective, which is the same as proving that

$H^0(\Gamma, V^*) \to H^0(\Gamma, V^*/V) \quad\quad\quad\quad\quad\quad\quad (1)$

is surjective.

It is known that one may use the set of 'multicurves' on $\Sigma$ in case (1a), whereas the integral homology of $\Sigma$, 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 $\Gamma$-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 $v-\gamma v\in V$ for every $\gamma\in\Gamma$), then $v$ is actually almost equal to an invariant element of $V^*$ (in the sense that there exists $w\in V$ such that $v-w\in H^0(\Gamma, V^*)$).