Automorphisms of $p$-local compact groups

TL;DR

This paper explores the automorphisms of p-local compact groups by constructing an explicit link between algebraic automorphisms of a related group and homotopy classes of self-equivalences of the classifying space, providing a more concrete understanding.

Contribution

It introduces a construction using Robinson's amalgam to explicitly relate algebraic automorphisms of a group to topological self-equivalences of the classifying space of a p-local compact group.

Findings

Established a split epimorphism from outer automorphisms of G to self homotopy equivalences

Provided an explicit algebraic description of automorphisms of classifying spaces

Connected algebraic and topological automorphism groups in the p-local setting

Abstract

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 a construction of Robinson of an amalgam $G$, realizing 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.