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The Lusternik-Schnirelmann category of a Lie groupoid


Author: Hellen Colman
Journal: Trans. Amer. Math. Soc. 362 (2010), 5529-5567
MSC (2010): Primary 22A22, 55M30, 18D05
DOI: https://doi.org/10.1090/S0002-9947-2010-05168-2
Published electronically: May 20, 2010
MathSciNet review: 2657690
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Abstract: We propose a new homotopy invariant for Lie groupoids which generalizes the classical Lusternik-Schnirelmann category for topological spaces. We use a bicategorical approach to develop a notion of contraction in this context. We propose a notion of homotopy between generalized maps given by the 2-arrows in a certain bicategory of fractions. This notion is invariant under Morita equivalence. Thus, when the groupoid defines an orbifold, we have a well-defined LS-category for orbifolds. We prove an orbifold version of the classical Lusternik-Schnirelmann theorem for critical points.


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  • 1. A. Adem, J. Leida and Y. Ruan, Orbifolds and Stringy Topology. Cambridge Tracts in Mathematics, Volume 171, 2007. MR 2359514 (2009a:57044)
  • 2. A. Adem and Y. Ruan, Twisted orbifold $ K$-theory. Comm. Math. Phys. 237 (2003), no. 3, 533-556. MR 1993337 (2004e:19004)
  • 3. M. F. Atiyah and G. Segal, On equivariant Euler characteristics. Journal of Geometry and Physics, Vol. 6 (1989), no. 4, 671-677 MR 1076708 (92c:19005)
  • 4. J. Baez and J. Dolan, From finite sets to Feynman diagrams. Mathematics unlimited--2001 and beyond, 29-50, Springer, Berlin, 2001. MR 1852152
  • 5. J. Benabou, Introduction to bicategories. In Midwest Category Seminar, volume 42 of Lecture Notes in Math., pages 1-77. Springer, 1967. MR 0220789 (36:3841)
  • 6. M. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften, 319. Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
  • 7. R. Brown, Topology and groupoids, BookSurge LLC, S. Carolina, 2006. MR 2273730
  • 8. H. Colman, LS-categories for foliated manifolds. Proceedings of the conference Foliations: Geometry and dynamics, Warsaw, 2000, World Scientific Publishing, 17-28. MR 1882763 (2002m:55007)
  • 9. H. Colman, Lusternik-Schnirelmann category of Orbifolds, Conference on Pure and Applied Topology, Isle of Skye, Scotland, 2005. available at http://www.abdn.ac.uk/$ \sim$wpe006/conference/skye/booklet.pdf
  • 10. H. Colman, On the 1-homotopy type of Lie groupoids, arXiv:math/0612257
  • 11. H. Colman and E. Macias, Tangential Lusternik-Schnirelmann category of foliations, J. London Math. Soc. 67 (2002), 745-756. MR 1895745 (2003a:57050)
  • 12. H. Colman and E. Macias, The transverse Lusternik-Schnirelmann category of a foliated manifold, Topology 40 (2) (2000), 419-430.
  • 13. O. Cornea, G. Lupton, J. Oprea and D. Tanré, Lusternik-Schnirelmann category, Mathematical Surveys and Monographs 103, American Mathematical Society, 2003. MR 1990857 (2004e:55001)
  • 14. M. Clapp and D. Puppe, Invariants of the Lusternik-Schnirelmann type and the topology of critical sets, Trans. Amer. Math. Soc. 298 (1986), 603-620. MR 860382 (88d:55004)
  • 15. L. Dixon, J. Harvey, C. Vafa and E. Witten, Strings on orbifolds, I, Nucl.Phys. B261(1985), 678-686. MR 818423 (87k:81104a)
  • 16. E. Fadell, The equivariant Ljusternik-Schnirelmann method for invariant functionals and relative cohomological index theories, In Méthodes Topologiques en Analyse Non Linéaire, ed. A. Granas, Montreal, 1985. MR 801933 (87b:58018)
  • 17. A. Haefliger, Homotopy and integrability (Manifolds, Amsterdam, 1970), Lectures Notes in Math. 197, Springer (1971), 133-163. MR 0285027 (44:2251)
  • 18. A. Haefliger, Groupoïdes d'holonomie et classifiants, Astérisque 116 (1984), 70-97. MR 755163 (86c:57026a)
  • 19. A. Haefliger, Orbi-espaces, in Sur les Groupes Hyperboliques d'après Mikhael Gromov, Progress in Mathematics, Birkhäuser 83 (1990), 203-213. MR 1086659
  • 20. I.M. James, Lusternik-Schnirelmann Category. Handbook of Algebraic Topology, Elsevier Science, Amsterdam, 1995, 1293-1310. MR 1361912 (97a:55003)
  • 21. I.M. James and J.R. Morris, Fibrewise category. Proceedings of the Royal Society of Edinburgh 119A (1991), 177-190. MR 1130605 (92g:55005)
  • 22. N. P. Landsman, Bicategories of operator algebras and Poisson manifolds, Fields Institute Communications 30 (2001), 271-286. MR 1867561 (2002h:46099)
  • 23. N. P. Landsman, Quantized reduction as a tensor product. Quantization of singular symplectic quotients, Progress in Mathematics, Birkhäuser 198 (2001), 137-180. MR 1938555 (2004c:53138)
  • 24. T. Leinster, The Euler characteristic of a category, Doc. Math. 13 (2008), 21-49. MR 2393085
  • 25. E. Lupercio and B. Uribe, Loop groupoids, gerbes, and twisted sectors on orbifolds, in Orbifolds in Mathematics and Physics, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163-184. MR 1950946 (2004c:58043)
  • 26. L. Lusternik and L. Schnirelmann, Méthodes topologiques dans les Problèmes Variationnels, Hermann, Paris, 1934.
  • 27. K. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, Lecture Notes Series 213, Cambridge University Press, 2005. MR 2157566 (2006k:58035)
  • 28. S. MacLane, Categories for the Working Mathematician, Springer-Verlag, 1971. MR 0354798 (50:7275)
  • 29. M. Mather, Pull-backs in homotopy theory. Canad. J. Math. 28 (1976), no. 2, 225-263. MR 0402694 (53:6510)
  • 30. I. Moerdijk, The classifying topos of a continuous groupoid, I, Transactions of the American Mathematical Society 310 (1988), 629-668. MR 973173 (90a:18005)
  • 31. I. Moerdijk, Orbifolds as groupoids, in Orbifolds in mathematics and physics, Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 205-222. MR 1950948 (2004c:22003)
  • 32. I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, 91. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2003. MR 2012261 (2005c:58039)
  • 33. I. Moerdijk and D. A. Pronk, Orbifolds, Sheaves and Groupoids, K-theory 13 (1997), 3-21. MR 1466622 (98i:22004)
  • 34. J. Mrčun, Stability and Invariants of Hilsum-Skandalis Maps, Ph.D. Thesis, Univ. of Utrecht (1996).
  • 35. D. A. Pronk, Etendues and stacks as bicategories of fractions, Compositio Mathematica 102 no. 3 (1996), 243-303. MR 1401424 (97d:18011)
  • 36. I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. 42 (1956), 359-363. MR 0079769 (18:144a)

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Additional Information

Hellen Colman
Affiliation: Department of Mathematics, Wilbur Wright College, 4300 N. Narragansett Avenue, Chicago, Illinois 60634
Email: hcolman@ccc.edu

DOI: https://doi.org/10.1090/S0002-9947-2010-05168-2
Received by editor(s): July 21, 2008
Received by editor(s) in revised form: July 14, 2009
Published electronically: May 20, 2010
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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