The homotopy theory of fusion systems

Broto, Carles; Levi, Ran; Oliver, Bob

doi:10.1090/S0894-0347-03-00434-X

The homotopy theory of fusion systems
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by Carles Broto, Ran Levi and Bob Oliver

J. Amer. Math. Soc. 16 (2003), 779-856

DOI: https://doi.org/10.1090/S0894-0347-03-00434-X

Published electronically: July 21, 2003

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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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