Cohomology of the hyperelliptic Torelli group

2011/10/03 by Tara Brendle, Brendle, Tara, Leah Childers +3
Mathematics · #20F36 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1110.0448

openalex publication_date 2011/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 H₁(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(S₃) 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.

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• Point pushing, homology, and the hyperelliptic involution
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