Classifying spaces for $1$–truncated compact Lie 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 ${Map}_{\ast }\left(BG,BH\right)$, $Map\left(BG,BH\right)$, and $Map\left(EG,B_{G}H\right)$ 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 when $H$ is compact abelian due to Lashof, May, and Segal.