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

## TL;DR

This paper investigates the homology of the commutator subgroup of the pure braid group, revealing infinite rank in certain homology groups and determining its cohomological dimension.

## Contribution

It provides new insights into the homological structure of the commutator subgroup of pure braid groups, including explicit calculations of homology and cohomological dimension.

## Key findings

- Homology groups contain free abelian groups of infinite rank for specified degrees.
- Cohomological dimension of the commutator subgroup is exactly n-2 for n ≥ 2.
- Homological properties are explicitly characterized for all n ≥ 2.

## 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 $1\leq l\leq n-2$. As a consequence we determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.