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Tangent spaces of bundles and of filtered diffeological spaces


Authors: J. Daniel Christensen and Enxin Wu
Journal: Proc. Amer. Math. Soc. 145 (2017), 2255-2270
MSC (2010): Primary 57P99, 58A05
DOI: https://doi.org/10.1090/proc/13334
Published electronically: January 11, 2017
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Abstract: We show that a diffeological bundle gives rise to an exact sequence of internal tangent spaces. We then introduce two new classes of diffeological spaces, which we call weakly filtered and filtered diffeological spaces, whose tangent spaces are easier to understand. These are the diffeological spaces whose categories of pointed plots are (weakly) filtered. We extend the exact sequence one step further in the case of a diffeological bundle with filtered total space and base space. We also show that the tangent bundle $ T^H X$ defined by Hector is a diffeological vector space over $ X$ when $ X$ is filtered or when $ X$ is a homogeneous space, and therefore agrees with the dvs tangent bundle introduced by the authors in a previous paper.


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  • [CSW] J. Daniel Christensen, Gordon Sinnamon, and Enxin Wu, The $ D$-topology for diffeological spaces, Pacific J. Math. 272 (2014), no. 1, 87-110. MR 3270173, https://doi.org/10.2140/pjm.2014.272.87
  • [CW1] J. Daniel Christensen and Enxin Wu, The homotopy theory of diffeological spaces, New York J. Math. 20 (2014), 1269-1303. MR 3312059
  • [CW2] J. Daniel Christensen and Enxin Wu, Tangent spaces and tangent bundles for diffeological spaces, Cahiers de Topologie et Geométrie Différentielle Catégoriques 57 (2016), no. 1, 3-50. Preprint available at http://arxiv.org/abs/1411.5425
  • [Do] P. Donato, Revêtement et groupe fondamental des espaces différentiels homogènes, Thèse de doctorat d'état, L'Université de Provence, Marseille, 1984.
  • [He] G. Hector, Géométrie et topologie des espaces difféologiques, Analysis and geometry in foliated manifolds (Santiago de Compostela, 1994), World Sci. Publ., River Edge, NJ, 1995, pp. 55-80 (French, with English summary). MR 1414196
  • [I1] P. Iglesias-Zemmour, Fibrations difféologiques et homotopie, Thèse de Doctorat Es-sciences, L'Université de Provence, Marseille, 1985.
  • [I2] Patrick Iglesias-Zemmour, Diffeology, Mathematical Surveys and Monographs, vol. 185, American Mathematical Society, Providence, RI, 2013. MR 3025051
  • [IKZ] Patrick Iglesias, Yael Karshon, and Moshe Zadka, Orbifolds as diffeologies, Trans. Amer. Math. Soc. 362 (2010), no. 6, 2811-2831. MR 2592936, https://doi.org/10.1090/S0002-9947-10-05006-3
  • [So] J.-M. Souriau, Groupes différentiels de physique mathématique, South Rhone seminar on geometry, II (Lyon, 1983) Travaux en Cours, Hermann, Paris, 1984, pp. 73-119 (French). MR 753860
  • [Wu] Enxin Wu, Homological algebra for diffeological vector spaces, Homology Homotopy Appl. 17 (2015), no. 1, 339-376. MR 3350086, https://doi.org/10.4310/HHA.2015.v17.n1.a17

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Additional Information

J. Daniel Christensen
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada
Email: jdc@uwo.ca

Enxin Wu
Affiliation: DIANA Group, Faculty of Mathematics, University of Vienna, Austria
Email: enxin.wu@univie.ac.at

DOI: https://doi.org/10.1090/proc/13334
Keywords: Tangent space, tangent bundle, diffeological bundle
Received by editor(s): November 1, 2015
Published electronically: January 11, 2017
Communicated by: Varghese Mathai
Article copyright: © Copyright 2017 American Mathematical Society

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