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 → 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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The second author was partially supported by NNSF of China (No. 112530) and STU Scientific Research Foundation for Talents (No. 760179).
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Christensen, J.D., Wu, E. Smooth classifying spaces. Isr. J. Math. 241, 911–954 (2021). https://doi.org/10.1007/s11856-021-2120-6
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DOI: https://doi.org/10.1007/s11856-021-2120-6