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
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:
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 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∗) is injective, which is the same as proving that
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 v−γv∈V for every γ∈Γ), then v is actually almost equal to an invariant element of V∗ (in the sense that there exists w∈V such that v−w∈H0(Γ,V∗)).