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

Ian J. Leary 

Faculty of Mathematical Studies, University of Southampton, 
Southampton SO17 1BJ, United Kingdom.  

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.  

Proceedings of the 1994 Barcelona Conference on Algebraic Topology,
Progress in Mathematics 136, Birkhaeuser (1996) 255-260.  

For a long time, I believed that the upper bound given for the
exponent of all but finitely many integral cohomology groups is always
attained.  A counterexample appears in my paper on cohomology of 
wreath products.