Cohomology of the hyperelliptic Torelli group

TL;DR

This paper investigates the cohomological properties of the hyperelliptic Torelli group, revealing its cohomological dimension, infinite generation of certain homology groups, and implications for the Burau representation kernel.

Contribution

It establishes the cohomological dimension of SI(S_g), shows the infinite generation of its top homology, and applies these results to the Burau representation kernel.

Findings

Cohomological dimension of SI(S_g) is g-1 for g > 0.

H_{g-1}(SI(S_g);Z) is infinitely generated for g > 1.

The kernel of the Burau representation at t=-1 has cohomological dimension floor(n/2).

Abstract

Let SI(S_g) denote the hyperelliptic Torelli group of a closed surface S_g of genus g. This is the subgroup of the mapping class group of S_g consisting of elements that act trivially on H_1(S_g;Z) and that commute with some fixed hyperelliptic involution of S_g. We prove that the cohomological dimension of SI(S_g) is g-1 when g > 0. We also show that H_g-1(SI(S_g);Z) is infinitely generated when g > 1. In particular, SI(S_3) is not finitely presentable. Finally, we apply our main results to show that the kernel of the Burau representation of the braid group B_n at t = -1 has cohomological dimension equal to the integer part of n/2, and it has infinitely generated homology in this top dimension.