Homotopy sequence of a topological groupoid with a basegroup and an obstruction to presentability of proper regular Lie groupoids. (English) Zbl 1329.22005

Topological groupoids are geometric objects representing singular quotient spaces (as foliations, equivalence relations, group actions, orbifolds, etc.).
The authors describe homotopy groups of \(K\)-pointed topological groupoids, where \(K\) is a topological group and there is a homomorphism from \(K\) to the groupoid. These homotopy groups are related to the ordinary homotopy groups through a long exact sequence. An obstruction to the presentability of proper regular Lie groupoids is given from the previous results.


22A22 Topological groupoids (including differentiable and Lie groupoids)
55Q05 Homotopy groups, general; sets of homotopy classes
58H05 Pseudogroups and differentiable groupoids
Full Text: DOI arXiv


[1] Carchedi, D.: Compactly generated stacks: a Cartesian closed theory of topological stacks. Adv. Math. 229(6), 3339-3397 (2012) · Zbl 1254.14003 · doi:10.1016/j.aim.2012.02.006
[2] Chen, W.: On a notion of maps between orbifolds. II. Homotopy and CW-complex. Commun. Contemp. Math. 8(6), 763-821 (2006) · Zbl 1108.22002 · doi:10.1142/S0219199706002283
[3] Conner, P.E., Floyd, E.E.: Differentiable Periodic Maps. Springer, Berlin (1964) · Zbl 0125.40103 · doi:10.1007/978-3-662-41633-4
[4] Crainic, M., Struchiner, I.: On the linearization theorem for proper Lie groupoids. arXiv:1103.5245v1 (2011) · Zbl 1286.53080
[5] Evens, S., Lu, J.-H., Weinstein, A.: Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Q. J. Math. Oxf. Ser. (2) 50(200), 417-436 (1999) · Zbl 0968.58014 · doi:10.1093/qjmath/50.200.417
[6] Haefliger, A.: Homotopy and integrability. In: Manifolds-Amsterdam 1970 (Proceedings of the Nuffic Summer School). Lecture Notes in Mathematics, vol. 197, pp. 133-163 (1971) · Zbl 0215.52403
[7] Haefliger, A.: Groupoïdes d’holonomie et classifiants. Transversal structure of foliations (Toulouse, 1982). Astérisque 116, 70-97 (1984) · Zbl 0562.57012
[8] Haefliger, A.: On the space of morphisms between étale groupoids. A celebration of the mathematical legacy of Raoul Bott. In: CRM Proceedings and Lecture Notes, vol. 50, pp. 139-150. American Mathematical Society, Providence (2010) · Zbl 1198.22001
[9] Henriques, A., Gepner, D.: Homotopy theory of orbispaces. arXiv:math/0701916v1 (2007)
[10] Jelenc, B.: Serre fibrations in the Morita category of topological groupoids. Topol. Appl. 160(1), 9-23 (2013) · Zbl 1269.22002 · doi:10.1016/j.topol.2012.09.013
[11] Lück, W., Oliver, B.: The completion theorem in \[KK\]-theory for proper actions of a discrete group. Topology 40(3), 585-616 (2001) · Zbl 0981.55002 · doi:10.1016/S0040-9383(99)00077-4
[12] Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids. Cambridge University Press, Cambridge (2005) · Zbl 1078.58011 · doi:10.1017/CBO9781107325883
[13] Moerdijk, I.: Classifying Spaces and Classifying Topoi. Springer, Berlin (1995) · Zbl 0838.55001
[14] Moerdijk, I.: Orbifolds as groupoids: an introduction. Orbifolds in mathematics and Physics (Madison 2001). Contemp. Math. 310, 205-222 (2002) · Zbl 1041.58009 · doi:10.1090/conm/310/05405
[15] Moerdijk, I.: Lie groupoids, gerbes, and non-abelian cohomology. K-Theory 28(3), 207-258 (2003) · Zbl 1042.58008
[16] Moerdijk, I., Mrčun, J.: Introduction to Foliations and Lie Groupoids. Cambridge University Press, Cambridge (2003) · Zbl 1029.58012 · doi:10.1017/CBO9780511615450
[17] Moerdijk, I., Mrčun, J.: Lie groupoids, sheaves and cohomology. Poisson geometry, deformation quantisation and group representations. Lond. Math. Soc. Lect. Note Ser. 323, 145-272 (2005) · Zbl 1082.58018
[18] Mrčun, J.: Stability and invariants of Hilsum-Skandalis maps. PhD thesis, Utrecht University. arXiv:math/0506484v1 (1996)
[19] Noohi, B.: Fibrations of topological stacks. arXiv:1010.1748v1 (2010) · Zbl 1237.57027
[20] Trentinaglia, G.: On the role of effective representations of Lie groupoids. Adv. Math. 225(2), 826-858 (2010) · Zbl 1216.58005 · doi:10.1016/j.aim.2010.03.014
[21] Weinstein, A.: Linearization of regular proper groupoids. J. Inst. Math. Jussieu 1(3), 493-511 (2002) · Zbl 1043.58009 · doi:10.1017/S1474748002000130
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.