Cohomology of the hyperelliptic Torelli group

Brendle, T. , Childers, L. and Margalit, D. (2013) Cohomology of the hyperelliptic Torelli group. Israel Journal of Mathematics, 195(2), pp. 613-630. (doi: 10.1007/s11856-012-0075-3)

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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 cohomo-logical dimension of SI(S g ) is g − 1 when g ≥ 1. We also show that H g−1(SI(S g ); ℤ) is infinitely generated when g ≥ 2. 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.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Brendle, Professor Tara
Authors: Brendle, T., Childers, L., and Margalit, D.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Israel Journal of Mathematics
ISSN (Online):1565-8511
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