A filtration of spectra arising from families of subgroups of symmetric groups

Abstract:

Let ${\mathcal F}_{n}$ be a family of subgroups of $\Sigma _{n}$ which is closed under taking subgroups and conjugates. Such a family has a classifying space, $B{\mathcal F}_{n}$, and we showed in an earlier paper that a compatible choice of ${\mathcal F}_{n}$ for each $n$ gives a simplicial monoid $\coprod _{n} B{\mathcal F}_{n}$, which group completes to an infinite loop space. In this paper we define a filtration of the associated spectrum whose filtration quotients, given an extra condition on the families, can be identified in terms of the classifying spaces of the families of subgroups that were chosen. This gives a way to go from group theoretic data about the families to homotopy theoretic information about the associated spectrum. We calculate two examples. The first is related to elementary abelian $p$-groups, and the second gives a new expression for the desuspension of $Sp^{m}(S^{0})/Sp^{m-1}(S^{0})$ as a suspension spectrum.