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The homotopy theory of fusion systems

Authors: Carles Broto, Ran Levi and Bob Oliver
Journal: J. Amer. Math. Soc. 16 (2003), 779-856
MSC (2000): Primary 55R35; Secondary 55R40, 20D20
Published electronically: July 21, 2003
MathSciNet review: 1992826
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Abstract: We define and characterize a class of $p$-complete spaces $X$ which have many of the same properties as the $p$-completions of classifying spaces of finite groups. For example, each such $X$ has a Sylow subgroup $BS\longrightarrow X$, maps $BQ\longrightarrow X$ for a $p$-group $Q$ are described via homomorphisms $Q\longrightarrow S$, and $H^*(X;\mathbb{F} _p)$ is isomorphic to a certain ring of ``stable elements'' in $H^*(BS;\mathbb{F} _p)$. These spaces arise as the ``classifying spaces'' of certain algebraic objects which we call ``$p$-local finite groups''. Such an object consists of a system of fusion data in $S$, as formalized by L. Puig, extended by some extra information carried in a category which allows rigidification of the fusion data.

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Additional Information

Carles Broto
Affiliation: Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain

Ran Levi
Affiliation: Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, United Kingdom

Bob Oliver
Affiliation: LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

Keywords: Classifying space, $p$-completion, finite groups, fusion.
Received by editor(s): August 3, 2001
Published electronically: July 21, 2003
Additional Notes: The first author is partially supported by MCYT grant BFM2001–2035
The second author is partially supported by EPSRC grant GR/M7831.
The third author is partially supported by UMR 7539 of the CNRS
All of the authors have been supported by EU grant HPRN-CT-1999-00119.
Article copyright: © Copyright 2003 American Mathematical Society