Extended moduli spaces and the Kan construction

Abstract

Let $Y$ be a CW-complex with a single 0-cell, let $K$ be its Kan group, a free simplicial group whose realization is a model for the space $\Omega Y$ of based loops on $Y$, and let $G$ be a Lie group, not necessarily connected. By means of simplicial techniques involving fundamental results of {\smc Kan's} and the standard $W$- and bar constructions, we obtain a weak $G$-equivariant homotopy equivalence from the geometric realization $|\roman{Hom}(K,G)|$ of the cosimplicial manifold $\roman{Hom}(K,G)$ of homomorphisms from $K$ to $G$ to the space $\roman{Map}^o(Y,BG)$ of based maps from $Y$ to the classifying space $BG$ of $G$ where $G$ acts on $BG$ by conjugation. Thus when $Y$ is a smooth manifold, the universal bundle on $BG$ being endowed with a universal connection, the space $|\roman{Hom}(K,G)|$ may be viewed as a model for the space of based gauge equivalence classes of connections on $Y$ for all topological types of $G$-bundles on $Y$ thereby yielding a rigorous approach to lattice gauge theory; this is illustrated in low dimensions.