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Lie groupoids determined by their orbit spaces (2310.11968v2)

Published 18 Oct 2023 in math.DG

Abstract: Given a Lie groupoid, we can form its orbit space, which carries a natural diffeology. More generally, we have a quotient functor from the Hilsum-Skandalis category of Lie groupoids to the category of diffeological spaces. We introduce the notion of a lift-complete Lie groupoid, and show that the quotient functor restricts to an equivalence of the categories: of lift-complete Lie groupoids with isomorphism classes of surjective submersive bibundles as arrows, and of quasi-\'{e}tale diffeological spaces with surjective local subductions as arrows. In particular, the Morita equivalence class of a lift-complete Lie groupoid, alternatively a lift-complete differentiable stack, is determined by its diffeological orbit space. Examples of lift-complete Lie groupoids include quasifold groupoids and \'{e}tale holonomy groupoids of Riemannian foliations.

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References (36)
  1. Alireza Ahmadi “Submersions, immersions, and étale maps in diffeology”, 2023 arXiv:2203.05994 [math.DG]
  2. J.A. Álvarez López and A. Candel “Equicontinuous foliated spaces” In Math. Z. 263.4, 2009, pp. 725–774 DOI: 10.1007/s00209-008-0432-4
  3. John C. Baez and Alexander E. Hoffnung “Convenient categories of smooth spaces” In Trans. Amer. Math. Soc. 363.11, 2011, pp. 5789–5825 DOI: 10.1090/S0002-9947-2011-05107-X
  4. Raymond Barre “De quelques aspects de la théorie des q𝑞qitalic_q-variétés différentielles et analytiques” In Ann. Inst. Fourier (Grenoble) 23.3, 1973, pp. 227–312 URL: http://www.numdam.org/item?id=AIF_1973__23_3_227_0
  5. “Differentiable stacks and gerbes” In J. Symplectic Geom. 9.3, 2011, pp. 285–341 URL: http://projecteuclid.org/euclid.jsg/1310388899
  6. Christian Blohmann “Stacky Lie groups” In Int. Math. Res. Not. IMRN, 2008, pp. Art. ID rnn 082\bibrangessep51 DOI: 10.1093/imrn/rnn082
  7. Henrique Bursztyn, Francesco Noseda and Chenchang Zhu “Principal actions of stacky Lie groupoids” In Int. Math. Res. Not. IMRN, 2020, pp. 5055–5125 DOI: 10.1093/imrn/rny142
  8. Domenico P.L. Castrigiano and Sandra A. Hayes “Orbits of Lie Group Actions are Weakly Embedded”, 2000 arXiv:math/0011241 [math.DG]
  9. Kuo Tsai Chen “Iterated path integrals” In Bull. Amer. Math. Soc. 83.5, 1977, pp. 831–879 DOI: 10.1090/S0002-9904-1977-14320-6
  10. Ryszard Engelking “General topology” Translated from the Polish by the author 6, Sigma Series in Pure Mathematics Heldermann Verlag, Berlin, 1989, pp. viii+529
  11. “Hausdorff Morita equivalence of singular foliations” In Ann. Global Anal. Geom. 55.1, 2019, pp. 99–132 DOI: 10.1007/s10455-018-9620-6
  12. A. Haefliger “Pseudogroups of local isometries” In Differential geometry (Santiago de Compostela, 1984) 131, Res. Notes in Math. Pitman, Boston, MA, 1985, pp. 174–197
  13. “Morphismes K𝐾Kitalic_K-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes)” In Ann. Sci. École Norm. Sup. (4) 20.3, 1987, pp. 325–390 URL: http://www.numdam.org/item?id=ASENS_1987_4_20_3_325_0
  14. Matias Hoyo and Rui Loja Fernandes “Riemannian metrics on differentiable stacks” In Math. Z. 292.1-2, 2019, pp. 103–132 DOI: 10.1007/s00209-018-2154-6
  15. Patrick Iglesias, Yael Karshon and Moshe Zadka “Orbifolds as diffeologies” In Trans. Amer. Math. Soc. 362.6, 2010, pp. 2811–2831 DOI: 10.1090/S0002-9947-10-05006-3
  16. Patrick Iglesias-Zemmour “Diffeology” 185, Mathematical Surveys and Monographs American Mathematical Society, Providence, RI, 2013, pp. xxiv+439 DOI: 10.1090/surv/185
  17. “Quasifolds, diffeology and noncommutative geometry” In J. Noncommut. Geom. 15.2, 2021, pp. 735–759 DOI: 10.4171/jncg/419
  18. “Quasifold groupoids and diffeological quasifolds” To appear in Transformation Groups, 2022 arXiv:2206.14776 [math.DG]
  19. Eugene Lerman “Orbifolds as stacks?” In Enseign. Math. (2) 56.3-4, 2010, pp. 315–363 DOI: 10.4171/LEM/56-3-4
  20. Gaël Meigniez “Prolongement des homotopies, Q-variétés et cycles tangents” In Ann. Inst. Fourier (Grenoble) 47.3, 1997, pp. 945–965 URL: http://www.numdam.org/item?id=AIF_1997__47_3_945_0
  21. Peter W. Michor and Cornelia Vizman “n𝑛nitalic_n-transitivity of certain diffeomorphism groups” In Acta Math. Univ. Comenian. (N.S.) 63.2, 1994, pp. 221–225
  22. David Miyamoto “Singular foliations through diffeology” Preprint. arXiv:2303.07494. To appear in Contemporary Mathematics (proceedings) for the special session ”Recent advances on diffeologies and their applications” held at the AMS-SMF-EMS Joint International Meeting in Genoble, July 2022., 2023 arXiv:2303.07494 [math.DG]
  23. David Miyamoto “The Basic de Rham Complex of a Singular Foliation” In Int. Math. Res. Not. IMRN, 2023, pp. 6364–6401 DOI: 10.1093/imrn/rnac044
  24. “Introduction to foliations and Lie groupoids” 91, Cambridge Studies in Advanced Mathematics Cambridge University Press, Cambridge, 2003, pp. x+173 DOI: 10.1017/CBO9780511615450
  25. Pierre Molino “Riemannian foliations” Translated from the French by Grant Cairns, With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu 73, Progress in Mathematics Birkhäuser Boston, Inc., Boston, MA, 1988, pp. xii+339 DOI: 10.1007/978-1-4684-8670-4
  26. Peter J. Olver “Applications of Lie groups to differential equations” 107, Graduate Texts in Mathematics Springer-Verlag, New York, 1993, pp. xxviii+513 DOI: 10.1007/978-1-4612-4350-2
  27. Elisa Prato “Sur une généralisation de la notion de V𝑉Vitalic_V-variété” In C. R. Acad. Sci. Paris Sér. I Math. 328.10, 1999, pp. 887–890 DOI: 10.1016/S0764-4442(99)80291-2
  28. Elisa Prato “Simple non-rational convex polytopes via symplectic geometry” In Topology 40.5, 2001, pp. 961–975 DOI: 10.1016/S0040-9383(00)00006-9
  29. I. Satake “On a generalization of the notion of manifold” In Proc. Nat. Acad. Sci. U.S.A. 42, 1956, pp. 359–363 DOI: 10.1073/pnas.42.6.359
  30. Ichirô Satake “The Gauss-Bonnet theorem for V𝑉Vitalic_V-manifolds” In J. Math. Soc. Japan 9, 1957, pp. 464–492 DOI: 10.2969/jmsj/00940464
  31. Jean-Marie Souriau “Groupes différentiels” In Differential geometrical methods in mathematical physics (Proc. Conf., Aix-en-Provence/Salamanca, 1979) 836, Lecture Notes in Math Springer, Berlin-New York, 1980, pp. pp 91–128
  32. Peter Stefan “Accessible sets, orbits, and foliations with singularities” In Proc. London Math. Soc. (3) 29, 1974, pp. 699–713 DOI: 10.1112/plms/s3-29.4.699
  33. Héctor J. Sussmann “Orbits of families of vector fields and integrability of distributions” In Trans. Amer. Math. Soc. 180, 1973, pp. 171–188 DOI: 10.2307/1996660
  34. Joel Villatoro “On the integrability of Lie algebroids by diffeological spaces”, 2023 arXiv:2309.07258 [math.DG]
  35. Jordan Watts “The differential structure of an orbifold” In Rocky Mountain J. Math. 47.1, 2017, pp. 289–327 DOI: 10.1216/RMJ-2017-47-1-289
  36. Jordan Watts “The orbit space and basic forms of a proper Lie groupoid” In Current trends in analysis, its applications and computation, Trends Math. Res. Perspect. Birkhäuser/Springer, Cham, 2022, pp. 513–523 DOI: 10.1007/978-3-030-87502-2“˙52

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