A bound on the exponent of the cohomology of BC-bundles

TL;DR

This paper establishes bounds on the exponent of cohomology in BC-bundles, proving key theorems about p-groups and analyzing torsion in cohomology of infinite groups, with implications for group theory and topology.

Contribution

It provides new bounds on cohomology exponents, proves a characterization of elementary abelian p-groups, and analyzes torsion in infinite groups' cohomology.

Findings

Proved a lower bound for cohomology exponents in BC-bundles.

Confirmed that p-groups are elementary abelian iff their cohomology has exponent p.

Identified torsion of order greater than the l.c.m. in Tate-Farrell cohomology of certain infinite groups.

Abstract

We give a lower bound for the exponent of certain elements in the integral cohomology of the total spaces of principal BC-bundles for C a finite cyclic group. As applications we give a proof of the theorem of A. Adem and H.-W. Henn that a p-group is elementary abelian if and only if its integral cohomology has exponent p, and we exhibit some infinite groups of finite virtual cohomological dimension whose Tate-Farrell cohomology contains torsion of order greater than the l.c.m. of the orders of their finite subgroups. We also give an upper bound for the exponent of all but finitely many of the integral cohomology groups of a finite group, in terms of the permutation representations of the group.