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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Kathryn Lesh
Journal: Trans. Amer. Math. Soc. 352 (2000), 3211-3237
MSC (2000): Primary 55P47; Secondary 55N20, 55P42
Published electronically: March 15, 2000
MathSciNet review: 1707701
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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.


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Additional Information

Kathryn Lesh
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390
Email: klesh@uoft02.utoledo.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02610-6
PII: S 0002-9947(00)02610-6
Received by editor(s): November 26, 1997
Published electronically: March 15, 2000
Article copyright: © Copyright 2000 American Mathematical Society