AUTOMORPHISMS OF p-LOCAL COMPACT GROUPS

Self equivalences of classifying spaces of p-local compact groups are well understood by means of the algebraic structure that gives rise to them, but explicit descriptions are lacking. In this paper we use Robinson’s construction of an amalgam G, realising a given fusion system, to produce a split epimorphism from the outer automorphism group of G to the group of homotopy classes of self homotopy equivalences of the classifying space of the corresponding p-local compact group. A p-local compact group is an algebraic object which is modelled on the p-local homotopy theory of classifying spaces of compact Lie groups and p-compact groups. These objects were constructed by Broto, Levi and Oliver in [BLO2, BLO3], and have been studied in an increasing number of papers since. Many aspects of the theory are still wide open, and among such aspects, a reasonable notion of maps between p-local compact groups remains quite elusive. Self equivalences of p-local compact groups are relatively well understood in terms of certain automorphisms of the underlying Sylow subgroup, but computing these groups of automorphisms is generally out of reach. This paper offers a way of regarding such automorphisms as being induced, in the appropriate sense, by automorphisms of certain discrete groups associated to that, in the finite case, were introduced by G. Robinson in [R]. The construction will be generalised here to the compact analog. We will refer to such groups as Robinson groups Before stating our results, we briefly recall the terminology, with a more detailed discussion in the next section. A p-local compact group is a triple G = (S,F ,L), where S is a discrete p-toral group (to be defined later), and F and L are categories whose objects are certain subgroups of S. The classifying space of G, which we denote by BG is the p-completed nerve |L|p . For any space X, let Out(X) denote the group of homotopy classes of self homotopy equivalences of X. We are now ready to state our main result. Theorem A. Let G = (S,F ,L) be a p-local compact group. Then there exist a group G, with a map μ : BG→ BG, and a a split epimorphism ω : Out(G)→ Out(BG). Furthermore, the following statements hold: (i) If [Ψ] ∈ Out(BG) and [Φ] ∈ ω−1([Ψ]) are represented by Ψ and Φ respectively, then μ ◦BΦ ' Ψ ◦ μ. (ii) Let Aut(G, 1S) ≤ Aut(G) denote the subgroup of automorphisms which restrict to the identity on S, and let Out(G, 1S) denote its quotient by the subgroup of inner automorphisms induced by conjugation by elements of CG(S). Then Ker(ω) ∼= Out(G, 1S) if p 6= 2, and for p = 2 there is a short exact sequence, 1→ Ker(ω)→ Out(G, 1S)→ lim ←− 1