Cohomological detection and regular elements in group cohomology

Abstract:

Suppose that $G$ is a compact Lie group or a discrete group of finite virtual cohomological dimension and that $k$ is a field of characteristic $p>0$. Suppose that $\mathcal {O}$ is a set of elementary abelian $p$-subgroups $G$ such that the cohomology $H^*(BG,k)$ is detected on the centralizers of the elements of $\mathcal {O}$. Assume also that $\mathcal {O}$ is closed under conjugation and that $E$ is in $\mathcal {O}$ whenever some subgroup of $E$ is in $\mathcal {O}$. Then there exists a regular element $\zeta$ in the cohomology ring $H^*(BG,k)$ such that the restriction of $\zeta$ to an elementary abelian $p$-subgroup $E$ is not nilpotent if and only if $E$ is in $\mathcal {O}$. The converse of the result is a theorem of Lannes, Schwartz and the second author. The results have several implications for the depth and associated primes of the cohomology rings.