Discrete models for thep–local homotopy theory of compact Lie groups andp–compact groups

Fusion systems

“…The main difference is that an extra "continuity" condition on the set of morphisms in the fusion system is needed. By analogy with the finite case, a classifying space of a saturated fusion system over a discrete p-toral group F is the p-completed realization |L| ∧ p of some associated linking system L. We refer to [BLO3, Definitions 2.2 and 4.1] for precise definitions, and to [BLO3] and [BLO5] for some of the properties of fusion and linking systems in this setting. More recently, Levi and Libman [LL] have proven the existence and uniqueness of linking systems (and hence classifying spaces) associated to saturated fusion systems over a discrete p-toral group.…”

“…∧ p for all such G. We refer to [BLO3,§ 9] for all of these results. In addition to compact Lie groups, saturated fusion systems over discrete p-toral groups and their classifying spaces can also be used to model p-compact groups, and p-completed classifying spaces of torsion linear groups (torsion subgroups of GL n (K) where K is a field and char(K) = p).…”

“…He was then able to use this to prove certain homotopy properties of the spaces |L| ∧ p (or |L|) which had not been shown in [BLO3], and which turned out to be much more difficult than in the finite case. For example, he showed that in such cases, H * (|L|; F p ) is isomorphic to the inverse limit (taken for P in F) of cohomology groups H * (P ; F p ), just as in the finite case.…”

“…For example, he showed that in such cases, H * (|L|; F p ) is isomorphic to the inverse limit (taken for P in F) of cohomology groups H * (P ; F p ), just as in the finite case. As a result, all of the properties of classifying spaces of saturated fusion systems over finite p-groups listed in Table 2 are now known in the discrete p-toral case; these are shown either in [BLO3] or in [Gz].…”

Unstable Adams operations acting onp–local compact groups and fixed points

“…The theory of p -local compact groups was introduced by C Broto, R Levi and B Oliver [7] as the natural generalization of p -local finite groups, also introduced by the same authors in [5], to include some infinite structures, such as compact Lie groups or p -compact groups, in an attempt to give categorical models for a larger class of p -completed classifying spaces.…”