×

Lie groupoids of mappings taking values in a Lie groupoid. (English) Zbl 07285968

The paper under review is a very significant contribution to the incipient theory of infinite-dimensional Lie groupoids modelled on locally convex spaces. Specifically, one performs a systematic investigation of Lie groupoid structures on spaces of functions defined on a manifold and taking values in a fixed finite-dimensional Lie groupoid. In particular, one explicitly describes the Lie algebroids associated to the infinite-dimensional Lie groupoids that are constructed in this way. Remarkably, in order to achieve their aims, the authors establish some tools that clearly hold an independent interest, such as composition operators on spaces of differentiable mappings, with emphasis on properties such as being submersion, immersion, étale, proper, and so on.

MSC:

22A22 Topological groupoids (including differentiable and Lie groupoids)
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
22E67 Loop groups and related constructions, group-theoretic treatment
46T10 Manifolds of mappings
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
58D15 Manifolds of mappings
58H05 Pseudogroups and differentiable groupoids
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Alzaareer, H.; Schmeding, A., Differentiable mappings on products with different degrees of differentiability in the two factors, Expo. Math. 33 (2015), no. 2, 184-222. MR 3342623 DOI: http://dx.doi.org/10.1016/j.exmath.2014.07.002 · Zbl 1330.46039 · doi:10.1016/j.exmath.2014.07.002
[2] Amiri, H., A group of continuous self-maps on a topological groupoid, Semigroup Forum (2017), 1-12. DOI: http://dx.doi.org/10.1007/s00233-017-9857-6 · Zbl 1393.22002 · doi:10.1007/s00233-017-9857-6
[3] Amiri, H.; Schmeding, A., Linking Lie groupoid representations and representations of infinite-dimensional Lie groups, 2018
[4] Amiri, H.; Schmeding, A., A differentiable monoid of smooth maps on Lie groupoids, J. Lie Theory 29 (4) (2019), 1167-1192 · Zbl 1436.58015
[5] Bastiani, A., Applications différentiables et variétés différentiables de dimension infinie, J. Analyse Math. 13 (1964), 1-114. MR 0177277 · Zbl 0196.44103 · doi:10.1007/BF02786619
[6] Beltiţă, D.; Goliński, T.; Jakimowicz, G.; Pelletier, F., Banach-Lie groupoids and generalized inversion, J. Funct. Anal. 276 (5) (2019), 1528-1574 · Zbl 1412.22008 · doi:10.1016/j.jfa.2018.12.002
[7] Bertram, W.; Glöckner, H.; Neeb, K.-H., Differential calculus over general base fields and rings, Expo. Math. 22 (2004), no. 3, 213-282. MR 2069671 (2005e:26042) DOI: http://dx.doi.org/10.1016/S0723-0869(04)80006-9 · Zbl 1099.58006 · doi:10.1016/S0723-0869(04)80006-9
[8] Bridson, M. R.; Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486 DOI: http://dx.doi.org/10.1007/978-3-662-12494-9 · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9
[9] Chen, Weimin, On a notion of maps between orbifolds. I. Function spaces, Commun. Contemp. Math. 8 (2006), no. 5, 569-620. MR 2263948 DOI: http://dx.doi.org/10.1142/S0219199706002246 · Zbl 1108.22003 · doi:10.1142/S0219199706002246
[10] Coufal, V.; Pronk, D.; Rovi, C.; Scull, L.; Thatcher, C., Orbispaces and their mapping spaces via groupoids: a categorical approach, Women in topology: collaborations in homotopy theory, Contemp. Math., vol. 641, Amer. Math. Soc., Providence, RI, 2015, pp. 135-166. MR 3380073 DOI: http://dx.doi.org/10.1090/conm/641/12857 · Zbl 1353.57025 · doi:10.1090/conm/641/12857
[11] Crainic, M.; Fernandes, R. L., Integrability of Lie brackets, Ann. of Math. (2) 157 (2003), no. 2, 575-620. MR 1973056 DOI: http://dx.doi.org/10.4007/annals.2003.157.575 · Zbl 1037.22003 · doi:10.4007/annals.2003.157.575
[12] Dahmen, R.; Glöckner, H.; Schmeding, A., Complexifications of infinite-dimensional manifolds and new constructions of infinite-dimensional Lie groups, 2016
[13] Eells, J. Jr., A setting for global analysis, Bull. Amer. Math. Soc. 72 (1966), 751-807. MR 0203742 DOI: http://dx.doi.org/10.1090/S0002-9904-1966-11558-6 · doi:10.1090/S0002-9904-1966-11558-6
[14] Engelking, R., General Topology, Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin, 1989 · Zbl 0684.54001
[15] Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions, Geometry and Analysis on Lie Groups (Strasburger, A., Hilgert, J., Neeb, K.-H., Wojtyński, W., eds.), Banach Center Publication, vol. 55, Warsaw, 2002, pp. 43-59 · Zbl 1020.58009
[16] Glöckner, H., Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal. 194 (2002), no. 2, 347-409. MR 1934608 DOI: http://dx.doi.org/10.1006/jfan.2002.3942 · Zbl 1022.22021 · doi:10.1006/jfan.2002.3942
[17] Glöckner, H., Regularity properties of infinite-dimensional Lie groups, and semiregularity, 2015
[18] Glöckner, H., Fundamentals of submersions and immersions between infinite-dimensional manifolds, 2016
[19] Glöckner, H.; Neeb, K.-H., Infinite-dimensional Lie groups, book in preparation · Zbl 1167.22013
[20] Glöckner, H.; Neeb, K.-H., Diffeomorphism groups of compact convex sets, Indag. Math. (N.S.) 28 (2017), no. 4, 760-783. MR 3679741 DOI: http://dx.doi.org/10.1016/j.indag.2017.04.004 · Zbl 1372.58008 · doi:10.1016/j.indag.2017.04.004
[21] Hamilton, R. S., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65-222. MR 656198 DOI: http://dx.doi.org/10.1090/S0273-0979-1982-15004-2 · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2
[22] Hjelle, E. O.; Schmeding, A., Strong topologies for spaces of smooth maps with infinite-dimensional target, Expo. Math. 35 (2017), no. 1, 13-53. MR 3626202 DOI: http://dx.doi.org/10.1016/j.exmath.2016.07.004 · Zbl 1379.58004 · doi:10.1016/j.exmath.2016.07.004
[23] Kac, V. G., Infinite-dimensional Lie algebras, third ed., Cambridge University Press, Cambridge, 1990. MR 1104219 DOI: http://dx.doi.org/10.1017/CBO9780511626234 · Zbl 0716.17022 · doi:10.1017/CBO9780511626234
[24] Keller, H. H., Differential calculus in locally convex spaces, Lecture Notes in Math., Vol. 417, Springer-Verlag, Berlin-New York, 1974. MR 0440592 (55 #13466) · Zbl 0293.58001
[25] Kriegl, A.; Michor, P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs 53, Amer. Math. Soc., Providence R.I., 1997 · Zbl 0889.58001
[26] Lang, S., Fundamentals of Differential Geometry, Graduate texts in mathematics 191, Springer, New York, \(^22001\) · Zbl 0995.53001
[27] Lerman, E., Orbifolds as stacks?, Enseign. Math. (2) 56 (2010), no. 3-4, 315-363. MR 2778793 DOI: http://dx.doi.org/10.4171/LEM/56-3-4 · Zbl 1221.14003 · doi:10.4171/LEM/56-3-4
[28] Meinrencken, E., Lie Groupoids and Lie algebroids, Lecture notes Fall 2017
[29] Meyer, R.; Zhu, Ch., Groupoids in categories with pretopology, Theory Appl. Categ. 30 (2015), Paper No. 55, 1906-1998. MR 3438234 · Zbl 1330.18005
[30] Michor, P. W., Manifolds of Differentiable Mappings, Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980. MR MR583436 (83g:58009) · Zbl 0433.58001
[31] Milnor, J., Remarks on infinite-dimensional Lie groups, Relativity, Groups and Topology, II (Les Houches, 1983), North-Holland, Amsterdam, 1984, pp. 1007-1057. MR MR830252 (87g:22024) · Zbl 0594.22009
[32] Moerdijk, I.; Mrčun, J., Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261 DOI: http://dx.doi.org/10.1017/CBO9780511615450 · Zbl 1029.58012 · doi:10.1017/CBO9780511615450
[33] Moerdijk, I.; Pronk, D. A., Orbifolds, sheaves and groupoids, \(K\)-Theory 12 (1997), no. 1, 3-21. MR 1466622 DOI: http://dx.doi.org/10.1023/A:1007767628271 · Zbl 0883.22005 · doi:10.1023/A:1007767628271
[34] Neeb, K.-H., Towards a Lie theory of locally convex groups, Japanese J. Math. 1 (2006), no. 2, 291-468. MR MR2261066 · Zbl 1161.22012 · doi:10.1007/s11537-006-0606-y
[35] Neeb, K.-H.; Wagemann, F., Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds, Geom. Dedicata 134 (2008), 17-60. MR 2399649 DOI: http://dx.doi.org/10.1007/s10711-008-9244-2 · Zbl 1143.22016 · doi:10.1007/s10711-008-9244-2
[36] Palais, R. S., Foundations of global non-linear analysis, W.A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0248880 · Zbl 0164.11102
[37] Palais, R. S., When proper maps are closed, Proc. Amer. Math. Soc. 24 (1970), 835-836. MR 0254818 DOI: http://dx.doi.org/10.2307/2037337 · Zbl 0189.53202 · doi:10.2307/2037337
[38] Pohl, A. D., The category of reduced orbifolds in local charts, J. Math. Soc. Japan 69 (2017), no. 2, 755-800. MR 3638284 DOI: http://dx.doi.org/10.2969/jmsj/06920755 · Zbl 1381.57016 · doi:10.2969/jmsj/06920755
[39] Pressley, A.; Segal, G., Loop groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986, Oxford Science Publications. MR 900587 · Zbl 0618.22011
[40] Roberts, D. M.; Schmeding, A., Extending Whitney’s extension theorem: nonlinear function spaces, to appear in Annales de l’Institut Fourier, 2018
[41] Roberts, D. M.; Vozzo, R. F., Smooth loop stacks of differentiable stacks and gerbes, Cah. Topol. Géom. Différ. Catég. 59 (2018), no. 2, 95-141. MR 3727316 · Zbl 1409.22003
[42] Roberts, D. M.; Vozzo, R. F., The smooth Hom-stack of an orbifold, 2016 MATRIX annals, MATRIX Book Ser., vol. 1, Springer, Cham, 2018, pp. 43-47. MR 3792515 · Zbl 1404.58031
[43] Schmeding, A., The diffeomorphism group of a non-compact orbifold, Dissertationes Math. (Rozprawy Mat.) 507 (2015), 179. MR 3328452 DOI: http://dx.doi.org/10.4064/dm507-0-1 · Zbl 1318.58005 · doi:10.4064/dm507-0-1
[44] Schmeding, A.; Wockel, Ch., The Lie group of bisections of a Lie groupoid, Ann. Global Anal. Geom. 48 (2015), no. 1, 87-123. MR 3351079 DOI: http://dx.doi.org/10.1007/s10455-015-9459-z · Zbl 1318.22002 · doi:10.1007/s10455-015-9459-z
[45] Schmeding, A.; Wockel, Ch., (Re)constructing Lie groupoids from their bisections and applications to prequantisation, Differential Geom. Appl. 49 (2016), 227-276. MR 3573833 DOI: http://dx.doi.org/10.1016/j.difgeo.2016.07.009 · Zbl 1358.58011 · doi:10.1016/j.difgeo.2016.07.009
[46] Weinmann, Th. O., Orbifolds in the framework of Lie groupoids, Ph.D. thesis, ETH Zürich, 2007. DOI: http://dx.doi.org/10.3929/ethz-a-005540169 · doi:10.3929/ethz-a-005540169
[47] Wittmann, J., The Banach manifold Ck(M,N), Differential Geom. Appl. 63 2) (2019), 166-185 · Zbl 1433.58012 · doi:10.1016/j.difgeo.2019.01.001
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.