The first cohomology of the mapping class group with coefficients in algebraic functions on the SL₂(C) moduli space

Abstract

Consider a compact surface of genus at least two. We prove that the first cohomology group of the mapping class group with coefficients in the space of algebraic functions on the SL 2 (C) moduli space vanishes.

Abelian Covers of Surfaces and the Homology of the Level L Mapping Class Group

“…Examining the formula for ω(·, ·) in Lemma 3.2, this immediately implies that ω( x v , y v ′ ) = 0, as desired. (2).…”

“…Consider ( x v , y, z L v ′ ) ∈ S g (2), so z can be realized by a simple closed nonseparating curve and {z} and {x, y} are strongly essentially disjoint. We can thus find subsurfaces X and X ′ of Σ g both of which contain the basepoint such that z ∈ Im(π 1 (X) → π 1 (Σ g )) and {x, y} ⊂ Im(π 1 (X ′ ) → π 1 (Σ g )).…”

“…In a somewhat different direction, a recent series of papers by Anderson and Villemoes [1,2,3] calculate the first homology groups of Mod n g,b with coefficients in certain spaces of functions on representations varieties of Mod n g,b .…”