Infinite Loop Spaces from Group Theory

                         Kathryn Lesh
                     University of Toledo

Abstract: We generalize the Barratt-Priddy-Quillen theorem  
\Omega B (\coprod B\Sigma_{n}) is homotopy equivalent to
QS^{0} by using tom Dieck's classifying spaces for a family 
of subgroups of a group.  We show how to take a compatible 
choice of families F_{n} of subgroups of \Sigma_{n} and 
obtain an infinite loop space by group completing 
\coprod BF_{n}. The spaces QS^{0} and K(Z,0) are recovered 
as extreme cases and the infinite loop spaces obtained from 
other families can be thought of as interpolating between 
stable homotopy and integral homology. We study two special 
cases: the family of subgroups of the alternating groups, 
and the family generated by elementary abelian p-subgroups 
whose generators are disjoint p-cycles. We compute the
infinite loop spaces which are formed in these cases and 
show that the latter is closely related to the cofiber of 
the transfer map.

This paper will appear in Mathematische Zeitschrift.