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Equivariant K-theory, groupoids and proper actions

Published online by Cambridge University Press:  21 November 2011

Jose Cantarero
Jose Cantarero
Affiliation:
Department of Mathematics, Stanford University, Stanford, California 94305USA. cantarer@stanford.edu
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Abstract

In this paper we define complex equivariant K-theory for actions of Lie groupoids using finite-dimensional vector bundles. For a Bredon-compatible Lie groupoid , this defines a periodic cohomology theory on the category of finite -CW-complexes. We also establish an analogue of the completion theorem of Atiyah and Segal. Some examples are discussed.

Keywords

K-theorygroupoidsproper actionscompletion theorem
Type
Research Article
Information
Journal of K-Theory , Volume 9 , Issue 3 , June 2012 , pp. 475 - 501
Copyright
Copyright © ISOPP 2011

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References

1.Adem, A., Leida, J. and Ruan, Y., Orbifolds and stringy topology, Cambridge Tracts in Mathematics 171, 2007.Google Scholar
2.Atiyah, M., Characters and cohomology of finite groups, Inst. Hautes Etudes Sci. Publ. Math. 9 (1961), 23–64.Google Scholar
3.Atiyah, M. and Hirzebruch, F., Vector bundles and homogeneous spaces, Proc. Sympos. Pure Math. 3, Amer. Math. Soc. 1961, 7–38.Google Scholar
4.Atiyah, M. and Segal, G., Equivariant K-theory and completion, J. Diff. Geom. 3 (1969), 1–18.Google Scholar
5.Atiyah, M. and Segal, G., Twisted K-theory, Ukr. Mat. Visn. 1 (2004), no. 3, 287–330.Google Scholar
6.Dwyer, C., Twisted equivariant K-theory for proper actions of discrete groups, Ph.D. Thesis, 2005.Google Scholar
7.Emerson, H., Meyer, R., Equivariant representable K-theory, J. Topol. 2 (2009), no. 1, 123–156.CrossRefGoogle Scholar
8.Freed, D., Hopkins, M., Teleman, C., Loop groups and twisted K-theory I, arXiv:math/0711.1906, 2007. To appear in J. Topology.Google Scholar
9.Freed, D., Hopkins, M., Teleman, C., Loop groups and twisted K-theory II, arXiv:math/0511232, 2005.Google Scholar
10.Freed, D., Hopkins, M., Teleman, C., Loop groups and twisted K-theory III, Annals of Math. 174 (2011), 947–1007.CrossRefGoogle Scholar
11.Freed, D., Hopkins, M., Teleman, C., Twisted equivariant K-theory with complex coefficients, J. Topol. 1 (2008), no. 1, 16–44.CrossRefGoogle Scholar
12.Gepner, D. and Henriques, A., Homotopy theory of orbispaces, arXiv:math.AT/0701916, 2007.Google Scholar
13.Kitchloo, N., Dominant K-theory and integrable highest weight representations of Kac-Moody groups, Advances in Math. 221 (2009), 1191–1126.CrossRefGoogle Scholar
14.Le Gall, P., Théorie de Kasparov équivariante et groupoïdes. I, K-Theory 16 (1999), no. 4, 361–390.Google Scholar
15.Lück, W., Survey on classifying spaces for families of subgroups, Infinite groups: geometric, combinatorial and dynamical aspects, Progr.Math. 248, 269–322, Birkhäuser, Basel, 2005.Google Scholar
16.Lück, W. and Oliver, B., The completion theorem in K-theory for proper actions of a discrete group, Topology 40 (2001), 585–616.Google Scholar
17.Moerdijk, I. and Mrcun, J., Introduction to foliations and Lie groupoids, Cambridge University Press, 2003.Google Scholar
18.Paterson, Alan L.T., Groupoids, inverse semigroups, and their operator algebras, Progress in mathematics 170, Birkhauser, Boston, 1998.Google Scholar
19.Phillips, N.C., Equivariant K-theory for proper actions, Pitman research notes in mathematics 178, 1989.Google Scholar
20.Phillips, N.C., Equivariant K-theory for proper actions II: Some cases in which finite dimensional bundles suffice, Index theory of elliptic operators, foliations and operator algebras, Contem. Math. 70 (1988), 205–227.Google Scholar
21.Sauer, J., K-theory for proper smooth actions of totally disconnected groups, Highdimensional manifold topology, World Sci. Publ., River Edge, NJ, 2003, 427–448.CrossRefGoogle Scholar
22.Segal, G., Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 105–112.Google Scholar
23.Segal, G., The representation ring of a compact Lie group, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 113–128.CrossRefGoogle Scholar
24.Segal, G., Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. 34 (1968), 129–151.CrossRefGoogle Scholar