A note on the homology of free abelianized extensions

Abstract

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