Article
MSC:
22A22,
22E65,
22E67,
46T10,
47H30,
58D15,
58H05 | MR 4188745 | Zbl 07285968 | DOI: 10.5817/AM2020-5-307
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Keywords:
Lie groupoid; Lie algebroid; topological groupoid; mapping groupoid; current groupoid; manifold of mappings; superposition operator; Nemytskii operator; pushforward; submersion; immersion; embedding; local diffeomorphism; \mathbbT1-cmd 1étale map; proper map; perfect map; orbifold groupoid; transitivity; local transitivity; local triviality; Stacey-Roberts Lemma
Lie groupoid; Lie algebroid; topological groupoid; mapping groupoid; current groupoid; manifold of mappings; superposition operator; Nemytskii operator; pushforward; submersion; immersion; embedding; local diffeomorphism; \mathbbT1-cmd 1étale map; proper map; perfect map; orbifold groupoid; transitivity; local transitivity; local triviality; Stacey-Roberts Lemma
Summary:
Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are prime examples of infinite-dimensional Lie groups modelled on locally convex spaces. In the present paper, we generalise this construction and show that differentiable mappings on a compact manifold (possibly with boundary) with values in a Lie groupoid form infinite-dimensional Lie groupoids which we call current groupoids. We then study basic differential geometry and Lie theory for these Lie groupoids of mappings. In particular, we show that certain Lie groupoid properties, like being a proper étale Lie groupoid, are inherited by the current groupoid. Furthermore, we identify the Lie algebroid of a current groupoid as a current algebroid (analogous to the current Lie algebra associated to a current Lie group). To establish these results, we study superposition operators \[ C^\ell (K,f)\colon C^\ell (K,M)\rightarrow C^\ell (K,N)\,,\;\, \gamma f\circ \gamma \] between manifolds of $C^\ell $-functions. Under natural hypotheses, $C^\ell (K,f)$ turns out to be a submersion (an immersion, an embedding, proper, resp., a local diffeomorphism) if so is the underlying map $f\colon M\rightarrow N$. These results are new in their generality and of independent interest.
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 DOI 10.1016/j.exmath.2014.07.002 | MR 3342623
[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 DOI 10.1007/s00233-017-9857-6 | MR 3750347
[3] Amiri, H., Schmeding, A.: Linking Lie groupoid representations and representations of infinite-dimensional Lie groups. 2018. MR 3951756
[4] Amiri, H., Schmeding, A.: A differentiable monoid of smooth maps on Lie groupoids. J. Lie Theory 29 (4) (2019), 1167–1192. MR 4022150
[5] Bastiani, A.: Applications différentiables et variétés différentiables de dimension infinie. J. Analyse Math. 13 (1964), 1–114. MR 0177277 DOI 10.1007/BF02786619 | MR 0177277
[6] Beltiţă, D., Goliński, T., Jakimowicz, G., Pelletier, F.: Banach-Lie groupoids and generalized inversion. J. Funct. Anal. 276 (5) (2019), 1528–1574. DOI 10.1016/j.jfa.2018.12.002 | MR 3912784
[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 DOI 10.1016/S0723-0869(04)80006-9 | MR 2069671 | Zbl 1099.58006
[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 DOI 10.1007/978-3-662-12494-9 | MR 1744486
[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 DOI 10.1142/S0219199706002246 | MR 2263948
[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 DOI 10.1090/conm/641/12857 | MR 3380073
[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 DOI 10.4007/annals.2003.157.575 | MR 1973056
[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 | MR 0203742
[14] Engelking, R.: General Topology. Sigma Series in Pure Mathematics, vol. 6, Heldermann, Berlin, 1989. MR 1039321 | 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. MR 1911979
[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 DOI 10.1006/jfan.2002.3942 | MR 1934608
[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.
[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 DOI 10.1016/j.indag.2017.04.004 | MR 3679741
[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 DOI 10.1090/S0273-0979-1982-15004-2 | MR 0656198 | Zbl 0499.58003
[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 DOI 10.1016/j.exmath.2016.07.004 | MR 3626202
[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 DOI 10.1017/CBO9780511626234 | MR 1104219
[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) MR 0440592 | 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. MR 1471480 | Zbl 0889.58001
[26] Lang, S.: Fundamentals of Differential Geometry. Graduate texts in mathematics 191, Springer, New York, $^2$2001. MR 1666820
[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 DOI 10.4171/LEM/56-3-4 | MR 2778793
[29] Meinrencken, E.: Lie Groupoids and Lie algebroids. Lecture notes Fall 2017.
[30] Meyer, R., Zhu, Ch.: Groupoids in categories with pretopology. Theory Appl. Categ. 30 (2015), Paper No. 55, 1906–1998. MR 3438234 MR 3438234
[31] Michor, P.W.: Manifolds of Differentiable Mappings. Shiva Mathematics Series, vol. 3, Shiva Publishing Ltd., Nantwich, 1980. MR MR583436 (83g:58009) MR 0583436 | Zbl 0433.58001
[32] 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) MR 0830252 | Zbl 0594.22009
[33] 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 DOI 10.1017/CBO9780511615450 | MR 2012261
[34] 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 DOI 10.1023/A:1007767628271 | MR 1466622
[35] Neeb, K.-H.: Towards a Lie theory of locally convex groups. Japanese J. Math. 1 (2006), no. 2, 291–468. MR MR2261066 DOI 10.1007/s11537-006-0606-y | MR 2261066 | Zbl 1161.22012
[36] 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 DOI 10.1007/s10711-008-9244-2 | MR 2399649 | Zbl 1143.22016
[37] Palais, R.S.: Foundations of global non-linear analysis. W.A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0248880 MR 0248880
[38] 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 DOI 10.2307/2037337 | MR 0254818
[39] 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 DOI 10.2969/jmsj/06920755 | MR 3638284
[40] Pressley, A., Segal, G.: Loop groups. Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1986, Oxford Science Publications. MR 900587 MR 0900587
[41] Roberts, D.M., Schmeding, A.: Extending Whitney’s extension theorem: nonlinear function spaces. to appear in Annales de l'Institut Fourier, 2018.
[42] 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 MR 3727316
[43] 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 MR 3792515
[44] 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 DOI 10.4064/dm507-0-1
[45] 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 DOI 10.1007/s10455-015-9459-z | MR 3351079
[46] 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 DOI 10.1016/j.difgeo.2016.07.009 | MR 3573833
[47] 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
[48] Wittmann, J.: The Banach manifold Ck(M,N). Differential Geom. Appl. 63 2) (2019), 166–185. DOI 10.1016/j.difgeo.2019.01.001 | MR 3903188
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