Homology of the braid group with coefficients in the reduced Burau representation

TL;DR

This paper computes the homology of the braid group with coefficients in the reduced Burau representation, revealing algebraic structures related to the action on cyclic covers of punctured discs.

Contribution

It provides an explicit calculation of the homology groups of the braid group with coefficients in the reduced Burau representation, a novel algebraic result.

Findings

Homology groups computed as modules over Laurent polynomial ring

Explicit algebraic structure of the homology revealed

Connections to cyclic covers of punctured discs established

Abstract

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 $n$-punctured disc. In this note, we calculate $H_*(B_n;V_n)$ as a module over the Laurent polynomial ring $\mathbb{Q}[t,t^{-1}]$.