The diffeology of Milnor's classifying space

TL;DR

This paper develops a diffeological framework for Milnor's classifying space, establishing classification theorems and connections for smooth principal bundles, with applications to infinite-dimensional groups and irrational tori.

Contribution

It introduces a diffeology on Milnor's classifying space, extending classical topological results to the diffeological setting and constructing diffeological connections.

Findings

Established a diffeology on Milnor's classifying space

Proved classification theorem for smooth principal bundles

Constructed diffeological connections on principal bundles

Abstract

We define a diffeology on the Milnor classifying space of a diffeological group $G$, constructed in a similar fashion to the topological version using an infinite join. Besides obtaining the expected classification theorem for smooth principal bundles, we prove the existence of a diffeological connection on any principal bundle (with mild conditions on the bundles and groups), and apply the theory to some examples, including some infinite-dimensional groups, as well as irrational tori.