Classifying spaces for 1-truncated compact Lie groups

TL;DR

This paper computes the homotopy types of various mapping spaces between classifying spaces of 1-truncated compact Lie groups, extending known results for finite and abelian cases and expressing them via homomorphism spaces.

Contribution

It provides explicit homotopy type calculations for mapping spaces between classifying spaces of 1-truncated compact Lie groups, generalizing previous finite and abelian cases.

Findings

Homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ are computed.

Results are expressed entirely in terms of spaces of homomorphisms from $G$ to $H$.

Generalizes known cases for finite and abelian groups.

Abstract

A 1-truncated compact Lie group is any extension of a finite group by a torus. In this note we compute the homotopy types of $Map_*(BG,BH)$, $Map(BG,BH)$, and $Map(EG, B_GH)^G$ for compact Lie groups $G$ and $H$ with $H$ 1-truncated, showing that they are computed entirely in terms of spaces of homomorphisms from $G$ to $H$. These results generalize the well-known case when $H$ is finite, and the case of $H$ compact abelian due to Lashof, May, and Segal.