Uniform exponent bounds for integral group homology

Abstract

We prove that, in each fixed degree, the exponent of the integral homology of a finite group is bounded solely in terms of the degree and the exponent of the group. The proof combines the solution of the restricted Burnside problem with a representability property of the bar construction and may be viewed as a torsion analogue of the method of acyclic models. We also use the Lyndon--Hochschild--Serre spectral sequence to obtain explicit bounds for finite solvable and nilpotent groups in terms of their derived length and nilpotency class.