Multiplicative structures and the twisted Baum-Connes assembly map
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- by Noé Bárcenas, Paulo Carrillo Rouse and Mario Velásquez PDF
- Trans. Amer. Math. Soc. 369 (2017), 5241-5269 Request permission
Abstract:
Using a combination of Atiyah-Segal ideas on one side and of Connes and Baum-Connes ideas on the other, we prove that the twisted geometric K-homology groups of a Lie groupoid have an external multiplicative structure extending hence the external product structures for proper cases considered by Adem-Ruan in 2003 or by Tu, Xu and Laurent-Gengoux in 2004. These twisted geometric K-homology groups are the left-hand sides of the twisted geometric Baum-Connes assembly maps recently constructed by Carrillo Rouse and Wang (2016), and hence one can transfer the multiplicative structure via the Baum-Connes map to the twisted K-theory groups whenever these assembly maps are isomorphisms.References
- Alejandro Adem and Yongbin Ruan, Twisted orbifold $K$-theory, Comm. Math. Phys. 237 (2003), no. 3, 533–556. MR 1993337, DOI 10.1007/s00220-003-0849-x
- Alejandro Adem, Yongbin Ruan, and Bin Zhang, A stringy product on twisted orbifold $K$-theory, Morfismos 11 (2007), no. 2, 33–64.
- Michael Atiyah and Graeme Segal, Twisted $K$-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330; English transl., Ukr. Math. Bull. 1 (2004), no. 3, 291–334. MR 2172633
- Noé Bárcenas, Jesús Espinoza, Michael Joachim, and Bernardo Uribe, Universal twist in equivariant $K$-theory for proper and discrete actions, Proc. Lond. Math. Soc. (3) 108 (2014), no. 5, 1313–1350. MR 3214681, DOI 10.1112/plms/pdt061
- Paul Baum, Nigel Higson, and Thomas Schick, A geometric description of equivariant $K$-homology for proper actions, Quanta of maths, Clay Math. Proc., vol. 11, Amer. Math. Soc., Providence, RI, 2010, pp. 1–22. MR 2732043, DOI 10.4310/PAMQ.2007.v3.n1.a1
- Paul Baum, Hervé Oyono-Oyono, Thomas Schick, and Michael Walter, Equivariant geometric $K$-homology for compact Lie group actions, Abh. Math. Semin. Univ. Hambg. 80 (2010), no. 2, 149–173. MR 2734682, DOI 10.1007/s12188-010-0034-z
- Alan L. Carey and Bai-Ling Wang, Fusion of symmetric D-branes and Verlinde rings, Comm. Math. Phys. 277 (2008), no. 3, 577–625. MR 2365446, DOI 10.1007/s00220-007-0399-8
- Paulo Carrillo Rouse, A Schwartz type algebra for the tangent groupoid, $K$-theory and noncommutative geometry, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2008, pp. 181–199. MR 2513337, DOI 10.4171/060-1/7
- Paulo Carrillo Rouse and Bai-Ling Wang, Twisted longitudinal index theorem for foliations and wrong way functoriality, Adv. Math. 226 (2011), no. 6, 4933–4986. MR 2775891, DOI 10.1016/j.aim.2010.12.026
- Paulo Carrillo Rouse and Bai-Ling Wang, Geometric Baum-Connes assembly map for twisted differentiable stacks, Ann. Sci. Éc. Norm. Supér. (4) 49 (2016), no. 2, 277–323 (English, with English and French summaries). MR 3481351, DOI 10.24033/asens.2283
- Matias del Hoyo and Rui Loja Fernandes, Riemannian metrics on Lie groupoids, arXiv preprint 1404.5989.
- Daniel S. Freed, Michael J. Hopkins, and Constantin Teleman, Loop groups and twisted $K$-theory I, J. Topol. 4 (2011), no. 4, 737–798. MR 2860342, DOI 10.1112/jtopol/jtr019
- Nigel Higson, A characterization of $KK$-theory, Pacific J. Math. 126 (1987), no. 2, 253–276. MR 869779, DOI 10.2140/pjm.1987.126.253
- Michel Hilsum and Georges Skandalis, Morphismes $K$-orientés d’espaces de feuilles et fonctorialité en théorie de Kasparov (d’après une conjecture d’A. Connes), Ann. Sci. École Norm. Sup. (4) 20 (1987), no. 3, 325–390 (French, with English summary). MR 925720, DOI 10.24033/asens.1537
- H. Blaine Lawson Jr. and Marie-Louise Michelsohn, Spin geometry, Princeton Mathematical Series, vol. 38, Princeton University Press, Princeton, NJ, 1989. MR 1031992
- Pierre-Yves Le Gall, Théorie de Kasparov équivariante et groupoïdes. I, $K$-Theory 16 (1999), no. 4, 361–390 (French, with English and French summaries). MR 1686846, DOI 10.1023/A:1007707525423
- I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
- El-kaïoum M. Moutuou, Equivariant $KK$-theory for generalised actions and Thom isomorphism in groupoid twisted $K$-theory, J. K-Theory 13 (2014), no. 1, 83–113. MR 3177819, DOI 10.1017/is013010018jkt244
- Janez Mrčun, Functoriality of the bimodule associated to a Hilsum-Skandalis map, $K$-Theory 18 (1999), no. 3, 235–253. MR 1722796, DOI 10.1023/A:1007773511327
- Markus J. Pflaum, Hessel Posthuma, and Xiang Tang, Geometry of orbit spaces of proper Lie groupoids, J. Reine Angew. Math. 694 (2014), 49–84. MR 3259039, DOI 10.1515/crelle-2012-0092
- Jean-Louis Tu, La conjecture de Baum-Connes pour les feuilletages moyennables, $K$-Theory 17 (1999), no. 3, 215–264 (French, with English and French summaries). MR 1703305, DOI 10.1023/A:1007744304422
- Jean-Louis Tu, The Baum-Connes conjecture for groupoids, $C^*$-algebras (Münster, 1999) Springer, Berlin, 2000, pp. 227–242. MR 1798599
- Jean-Louis Tu and Ping Xu, The ring structure for equivariant twisted $K$-theory, J. Reine Angew. Math. 635 (2009), 97–148. MR 2572256, DOI 10.1515/CRELLE.2009.077
- Jean-Louis Tu, Ping Xu, and Camille Laurent-Gengoux, Twisted $K$-theory of differentiable stacks, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 841–910 (English, with English and French summaries). MR 2119241, DOI 10.1016/j.ansens.2004.10.002
- Nguyen Tien Zung, Proper groupoids and momentum maps: linearization, affinity, and convexity, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 5, 841–869 (English, with English and French summaries). MR 2292634, DOI 10.1016/j.ansens.2006.09.002
Additional Information
- Noé Bárcenas
- Affiliation: Centro de Ciencias Matemáticas, UNAM, Ap. Postal 61-3 Xangari, Morelia, Michoacán, México 58089
- Email: barcenas@matmor.unam.mx
- Paulo Carrillo Rouse
- Affiliation: Institut de Mathématiques de Toulouse, 118, route de Narbonne, F-31062 Toulouse, France
- MR Author ID: 873936
- Email: paulo.carrillo@math.univ-toulouse.fr
- Mario Velásquez
- Affiliation: Departamento de Matemáticas, Pontificia Universidad Javeriana, Cra. 7, No. 43-82 - Edificio Carlos Ortíz 5to piso, Bogotá D.C, Colombia
- MR Author ID: 1031622
- Email: mavelasquezm@gmail.com
- Received by editor(s): March 31, 2016
- Received by editor(s) in revised form: July 11, 2016
- Published electronically: March 17, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 369 (2017), 5241-5269
- MSC (2010): Primary 19K56, 19L50; Secondary 46L80
- DOI: https://doi.org/10.1090/tran/7024
- MathSciNet review: 3632567