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

Bianchi, Andrea

doi:10.1090/proc/15404

On the homology of the commutator subgroup of the pure braid group
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Proc. Amer. Math. Soc. 149 (2021), 2387-2401 Request permission

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 $\mathrm {cd}([P_{n},P_{n}])=n-2$.

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