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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References
R. Andrade, Example of fiber bundle that is not a fibration, https://mathoverflow.net/q/126228.
N. Bourbaki, Topologie Générale. Ch. IX, Hermann, Paris, 1958.
J. D. Christensen, G. Sinnamon and E. Wu, The D-topology for diffeological spaces, Pacific Journal of Mathematics 272 (2014), 87–110.
J. D. Christensen and E. Wu, The homotopy theory of diffeological spaces, New York Journal of Mathematics 20 (2014), 1269–1303.
J. D. Christensen and E. Wu, Diffeological vector spaces, Pacific Journal of Mathematics 303 (2019), 73–92.
A. Dold, Partitions of unity in the theory of fibrations, Annals of Mathematics 78 (1963), 223–255.
T. Goodwillie, Non trivial vector bundle over non-paracompact contractible space, https://mathoverflow.net/q/106563.
D. Husemoller, Fibre Bundles, Graduate Texts in Mathematics, Vol. 20, Springer, New York, 1994.
P. Iglesias-Zemmour, Fibrations difféologiques et Homotopie, Thèse de Doctorat Es-sciences, L’Université de Provence, 1985.
P. Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs, Vol. 185, American Mathematical Society, Providence, RI, 2013.
A. Kriegl and P. W. Michor, The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Vol. 53, American Mathematical Society, Providence, RI, 1997.
J.-P. Magnot and J. Watts, The diffeology of Milnor’s classifying space, Topology and its Applications 232 (2017), 189–213.
J. Milnor, Construction of universal bundles. II, Annals of Mathematics 63 (1956), 430–436.
M. A. Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, Journal of Differential Geometry 14 (1979), 255–293.
G. Segal, Classifying spaces and spectral sequences, Institut des Hautes Études Scientifiques. Publications Mathématiques 34 (1968), 105–112.
J.-M. Souriau, Groupes différentiels, in Differential Geometrical Methods in Mathematical Physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979), Lecture Notes in Mathematics, Vol. 836, Springer, Berlin, 1980, pp. 91–128.
J.-M. Souriau, Groupes différentiels de physique mathématique, in South Rhone Seminar on Geometry, II (Lyon, 1983), Travaux en Cours, Hermann, Paris, 1984, pp. 73–119.
N. E. Steenrod, Milgram’s classifying space of a topological group, Topology 7 (1968), 349–368.
E. Wu, Homological algebra for diffeological vector spaces, Homology, Homotopy and Applications 17 (2015), 339–376.
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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