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RATIONAL LOCAL SYSTEMS AND CONNECTED FINITE LOOP SPACES

Part of: Spectral sequences Homotopy theory

Published online by Cambridge University Press:  14 January 2021

DREW HEARD
DREW HEARD*
Affiliation:
Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway e-mail:[email protected]
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Abstract

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Greenlees has conjectured that the rational stable equivariant homotopy category of a compact Lie group always has an algebraic model. Based on this idea, we show that the category of rational local systems on a connected finite loop space always has a simple algebraic model. When the loop space arises from a connected compact Lie group, this recovers a special case of a result of Pol and Williamson about rational cofree G-spectra. More generally, we show that if K is a closed subgroup of a compact Lie group G such that the Weyl group WGK is connected, then a certain category of rational G-spectra “at K” has an algebraic model. For example, when K is the trivial group, this is just the category of rational cofree G-spectra, and this recovers the aforementioned result. Throughout, we pay careful attention to the role of torsion and complete categories.

MSC classification

Primary: 55P91: Equivariant homotopy theory
Secondary: 55P35: Loop spaces 55T15: Adams spectral sequences
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 136 - 164
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

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