Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-28T03:11:53.619Z Has data issue: false hasContentIssue false
  • English
  • Français

A new description of equivariant cohomology for totally disconnected groups

Published online by Cambridge University Press:  11 February 2008

Christian Voigt
Christian Voigt
Affiliation:
cvoigt@math.uni-muenster.deMathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 6248149 MünsterGermany
Get access

Abstract

We consider smooth actions of totally disconnected groups on simplicial complexes and compare different equivariant cohomology groups associated to such actions. Our main result is that the bivariant equivariant cohomology theory introduced by Baum and Schneider can be described using equivariant periodic cyclic homology. This provides a new approach to the construction of Baum and Schneider as well as a computation of equivariant periodic cyclic homology for a natural class of examples. In addition we discuss the relation between cosheaf homology and equivariant Bredon homology. Since the theory of Baum and Schneider generalizes cosheaf homology we finally see that all these approaches to equivariant cohomology for totally disconnected groups are closely related.

Keywords

cyclic homologyequivariant cohomology
Type
Research Article
Information
Journal of K-Theory , Volume 1 , Issue 3 , June 2008 , pp. 431 - 472
Copyright
Copyright © ISOPP 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Baum, P., Connes, A., Geometric K-theory for Lie groups and foliations, Preprint IHES (1982)Google Scholar
2.Baum, P., Connes, A., Chern character for discrete groups, in: A fête of topology, 163–232, Academic Press, 1988CrossRefGoogle Scholar
3.Baum, P., Connes, A., Higson, N., Classifying space for proper actions and K-theory of group C*-algebras, in C*-algebras: 1943 – 1993 (San Antonio, TX, 1993), 241–291, Contemp. Math. 167, 1994Google Scholar
4.Baum, P., Schneider, P., Equivariant bivariant Chern character for profinite groups, K-theory 25 (2002), 313–353CrossRefGoogle Scholar
5.Bernstein, J., Lunts, V., Equivariant sheaves and functors, Lecture Notes in Mathematics 1578, Springer, 1994Google Scholar
6.Bernstein, J., Zelevinskii, A., Representations of the group GL(n,F) where F is a local non-archimedian field, Russian Math. Surveys 31 (1976), 1–68CrossRefGoogle Scholar
7.Bredon, G., Equivariant cohomology theories, Lecture Notes in Mathematics 34, Springer, 1967Google Scholar
8.Bredon, G., Sheaf theory, second edition, Graduate Texts in Mathematics 170, Springer, 1997Google Scholar
9.Connes, A., Noncommutative differential geometry, Publ. Math. IHES 39 (1985), 257–360Google Scholar
10.Connes, A., Noncommutative Geometry, Academic Press, 1994Google Scholar
11.Higson, N., Nistor, V., Cyclic homology of totally disconnected groups acting on buildings, J. Funct. Anal. 141 (1996) no. 2, 466–495CrossRefGoogle Scholar
12.Lück, W., Transformation groups and algebraic K-theory, Lecture Notes in Mathematics 1408, Springer, 1989Google Scholar
13.Lück, W., Chern characters for proper equivariant homology theories and applications to K- and L-theory, J. Reine Angew. Math. 543 (2002), 193–234Google Scholar
14.Meyer, R., Analytic cyclic cohomology, Preprintreihe SFB 478, Geometrische Strukturen in der Mathematik, Münster, 1999Google Scholar
15.Meyer, R., Smooth group representations on bornological vector spaces, Bull. Sci. Math. 128 (2004), 127–166CrossRefGoogle Scholar
16.Spanier, E. H., Algebraic Topology, McGraw-Hill, 1966Google Scholar
17.Teleman, N., Microlocalisation de l'homologie de Hochschild, C. R. Acad. Sci. Paris 326 (1998), 1261–1264CrossRefGoogle Scholar
18.Voigt, C., Equivariant cyclic homology, Preprintreihe SFB 478, Geometrische Strukturen in der Mathematik, Münster, 2003Google Scholar
19.Voigt, C., Equivariant periodic cyclic homology, Journal of the Inst. Math. Jussieu 6 (2007), 689–763CrossRefGoogle Scholar
20.Voigt, C., Chern character for totally disconnected groups, arXiv:math.KT/0608626 (2006)Google Scholar
21.Weibel, C. A., An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press, 1994Google Scholar
22.Willis, G. A., Totally disconnected groups and proofs of conjectures of Hofmann and Mukherjea, Bull. Austral. Math. Soc. 51 (1995), 489–494CrossRefGoogle Scholar
23.Wodzicki, M., The long exact sequence in cyclic homology associated with an extension of algebras, C. R. Acad. Sci. Paris 306 (1988), 399–403Google Scholar
24.Wodzicki, M., Excision in cyclic homology and in rational algebraic K-Theory, Ann. of Math. 129 (1989), 591–639CrossRefGoogle Scholar