A note on the homology of free abelianized extensions

Abstract:

Let $G$ be a group given by a free presentation $G = F/N$, and $N’$ the commutator subgroup of $N$. The quotient $F/N’$ is called a free abelianized extension of $G$. We study the integral homology of $F/N’$. In particular, if $G$ has no elements of order $p$ ($p$ an odd prime), we determine the $p$-torsion in dimension ${p^2}$ in terms of the modulo $p$ homology of $G$. This extends results of Kuz’min [5, 6] describing the $p$-torsion in smaller dimensions. Our approach is based on examining the homology of $G$ with coefficients in symmetric powers of the augmentation ideal, which we relate to the integral homology of $F/N’$.