Homology of braid groups, the Burau representation, and $\mathbb{F}_q$-points on superelliptic curves

Weiyan Chen

Published 2015-06-06Version 1

The (reduced) Burau representation $V_n$ of the braid group $B_n$ is obtained from the action of $B_n$ on the homology of an infinite cyclic cover of the disc with $n$ punctures. In this paper, we calculate $H_*(B_n;V_n)$. As an application, we show that the expected number of $\mathbb{F}_q$-points on a random superelliptic curve is equal to $q$.