On the homology of the commutator subgroup of the pure braid group

Abstract

We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free abelian group of infinite rank for all $1\le l\le n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\ge 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.

21 pages