A completion theorem for fusion systems

Abstract

We show that the twisted K-theory of the classifying space of a plocal finite group is isomorphic to the completion of the Grothendieck group of twisted representations of the fusion system with respect to the augmentation ideal of the representation ring of the fusion system. We use this result to compute the K-theory of the Ruiz-Viruel exotic 7-local finite groups.2010 Mathematics Subject Classification. 55R35 (primary), 19A22, 19L50, 20D20 (secondary).

Uniqueness of factorization for fusion-invariant representations

“…When F is clear, we also call them fusion-invariant representations. These representations have been studied in [4], [8], [12] and [23], but also in [7], [22], [27] and [28] in a more general context.…”

“…We also use the notation R G (S) when F = F S (G). In general, R(F ) is a free abelian group whose rank is the number of F -conjugacy classes of elements of S (see Corollary 2.2 in [4]). However, the monoid Rep(F ) is not free in general as the following example from [12] shows.…”

“…Note that this result can not be extended to order p 4 , as the non-factorial examples in [12] and [23] are given over Sylows of order 3 4 and 2 4 , respectively.…”

“…Given a p-local finite group (S, F , L), the Grothendieck group of complex vector bundles over |L| ∧ p was found in [8] to be isomorphic to the representation ring R(F ), which can expressed as an inverse limit of representation rings over the orbit category of F . On the other hand, the K-theory of |L| ∧ p is isomorphic to the completion of R(F ) with respect to the augmentation ideal (see [4]).…”

“…This result can not be taken further, since the examples without uniqueness of factorization in [12] and [23] correspond to fusion systems over groups of order 3 4 and 2 4 , respectively. They are in fact the fusion systems of Σ 9 and P SL 3 (F 3 ) at the primes 3 and 2.…”