Identification of infinite loop spaces arising from group theory
                            Kathryn Lesh

     Let 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, BF_{n}, and we showed in an earlier work that a
compatible choice of F_{n} for each n gives a simplicial monoid
\coprod_{n} BF_{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 quotients of symmetric powers of
spheres as suspension spectra.

Address: Department of Mathematics, University of Toledo, Toledo OH 43606

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