The homotopy theory of fusion systems

Broto, Carles¹ ; Levi, Ran² ; Oliver, Bob³

¹ Departament de Matemàtiques, Universitat Autònoma de Barcelona, E–08193 Bellaterra, Spain
² Department of Mathematical Sciences, University of Aberdeen, Meston Building 339, Aberdeen AB24 3UE, United Kingdom
³ LAGA, Institut Galilée, Av. J-B Clément, 93430 Villetaneuse, France

Journal of the American Mathematical Society, Tome 16 (2003) no. 4, pp. 779-856

Résumé

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.

Détail

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

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