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Equivariant properties of symmetric products


Author: Stefan Schwede
Journal: J. Amer. Math. Soc. 30 (2017), 673-711
MSC (2010): Primary 55P91
DOI: https://doi.org/10.1090/jams/879
Published electronically: February 24, 2017
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Abstract: The filtration of the infinite symmetric product of spheres by the number of factors provides a sequence of spectra between the sphere spectrum and the integral Eilenberg-Mac Lane spectrum. This filtration has received a lot of attention, and the subquotients are interesting stable homotopy types. While the symmetric product filtration has been a major focus of research since the 1980s, essentially nothing was known when one adds group actions into the picture. We investigate the equivariant stable homotopy types, for compact Lie groups, obtained from this filtration of infinite symmetric products of representation spheres. The situation differs from the non-equivariant case; for example, the subquotients of the filtration are no longer rationally trivial and on the zeroth equivariant homotopy groups an interesting filtration of the augmentation ideals of the Burnside rings arises. Our method is by global homotopy theory; i.e., we study the simultaneous behavior for all compact Lie groups at once.


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Additional Information

Stefan Schwede
Affiliation: Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
Email: schwede@math.uni-bonn.de

DOI: https://doi.org/10.1090/jams/879
Keywords: Symmetric product, compact Lie group, equivariant stable homotopy theory
Received by editor(s): March 17, 2014
Received by editor(s) in revised form: May 30, 2016
Published electronically: February 24, 2017
Article copyright: © Copyright 2017 American Mathematical Society