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

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. We are mainly interested in the case when the total space is BG for some discrete group G having a central subgroup isomorphic to C. 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. Our examples include a class of groups having similar properties discovered by Adem and J. Carlson. As a third application, we examine the integral cohomology of a class of p-groups expressible as central extensions with cyclic kernel and quotient abelian of p-rank two. For each such G we determine the minimal n such that almost all (i.e. all but possibly finitely many) of the groups H ⁱ(BG) have exponent dividing p ⁿ. The lemma we use to give an upper bound for the exponents of almost all of the groups H ⁱ(BG) applies to any p-group and may be of independent interest. Here, and throughout the paper, the coefficients for cohomology are to be the integers when not otherwise stated, and we write ℤ _n , for the integers modulo n. The author gratefully acknowledges that this work was funded by the DGICYT.