Smooth classifying spaces

J. Daniel Christensen; Enxin Wu

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arXiv · 1709.10517

Smooth classifying spaces

J. Daniel Christensen·Enxin Wu

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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.

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BibTeX RIS

J. Daniel Christensen, Enxin Wu. 2017-09-29. Smooth classifying spaces. https://arxiv.org/abs/1709.10517

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