The rational cohomology of the mapping class group vanishes in its virtual cohomological dimension

TL;DR

This paper proves that the rational cohomology of the mapping class group vanishes in its virtual cohomological dimension for genus g >= 2, using combinatorial methods and a theorem by Broaddus.

Contribution

It establishes the vanishing of the top rational cohomology of Mod_g at its virtual cohomological dimension, providing a new proof with combinatorial techniques.

Findings

H^{4g-5}(Mod_g; Q) = 0 for g >= 2

Uses combinatorics of chord diagrams and Broaddus's theorem

Advances understanding of the cohomological properties of mapping class groups

Abstract

Let Mod_g be the mapping class group of a genus g >= 2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^{4g-5}(Mod_g; Q) = 0.