Smooth classifying spaces

Abstract

We develop the theory of smooth principal bundles for a smooth group $G$, using the framework of diffeological spaces. After giving new examples showing why arbitrary principal bundles cannot be classified, we define $D$-numerable bundles, the smooth analogs of numerable bundles from topology, and prove that pulling back a $D$-numerable bundle along smoothly homotopic maps gives isomorphic pullbacks. We then define smooth structures on Milnor's spaces $EG$ and $BG$, show that $EG \to BG$ is a $D$-numerable principal bundle, and prove that it classifies all $D$-numerable principal bundles over any diffeological space. We deduce analogous classification results for $D$-numerable diffeological bundles and vector bundles.