The $p$-local homotopy theory of classifying spaces of finite and compact Lie groups is modelled algebraically by the very general concept of $p$-local finite groups and their “positive dimensional” analogs $p$-local compact groups. This setup allows one to study homotopy theoretic properties of classifying spaces, which are intrinsic to the $p$-local structure of the group in question, on one hand, and on the other hand expand the theory to a variety of exotic examples, understand common features of families of spaces in a general context, and explore new phenomena without having to resort to the group structure or geometric features of the group in question. This talk will give a specific example of how an interesting homotopy theoretic property of finite groups is a particular case of a much more general statement.

For a finite group $G$ the homology of $\Omega BG^\wedge_p$ with coefficients in a field of characteristics $p$ was shown by Benson to admit a definition in purely algebraic terms. We will present an approach which generalises Benson’s construction to $p$-local group theory. This is motivated in part by the aim of providing a tool to distinguish the family of $p$-compact groups from among all $p$-local compact groups.