# Cyclic homology and equivariant theories

Annales de l'institut Fourier (1987)

- Volume: 37, Issue: 4, page 15-28
- ISSN: 0373-0956

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topBrylinski, Jean-Luc. "Cyclic homology and equivariant theories." Annales de l'institut Fourier 37.4 (1987): 15-28. <http://eudml.org/doc/74771>.

@article{Brylinski1987,

abstract = {In this article, we present two possible extensions of the classical theory of equivariant cohomology. The first, due to P. Baum, R. MacPherson and the author, is called the “delocalized theory". We attempt to present it in very concrete form for a circle action on a smooth manifold. The second is the cyclic homology of the crossed- product algebra of the algebra of smooth functions on a manifold, by the convolution algebra of smooth functions on a Lie group, when such Lie group act on the manifold. In the case of a compact group, one recovers equivariant $K$-theory. One also obtains geometrically interesting results, for instance in the case of a discrete group.},

author = {Brylinski, Jean-Luc},

journal = {Annales de l'institut Fourier},

keywords = {smooth circle actions on manifolds; cyclic homology; -algebra; convolution algebra; equivariant K-theory},

language = {eng},

number = {4},

pages = {15-28},

publisher = {Association des Annales de l'Institut Fourier},

title = {Cyclic homology and equivariant theories},

url = {http://eudml.org/doc/74771},

volume = {37},

year = {1987},

}

TY - JOUR

AU - Brylinski, Jean-Luc

TI - Cyclic homology and equivariant theories

JO - Annales de l'institut Fourier

PY - 1987

PB - Association des Annales de l'Institut Fourier

VL - 37

IS - 4

SP - 15

EP - 28

AB - In this article, we present two possible extensions of the classical theory of equivariant cohomology. The first, due to P. Baum, R. MacPherson and the author, is called the “delocalized theory". We attempt to present it in very concrete form for a circle action on a smooth manifold. The second is the cyclic homology of the crossed- product algebra of the algebra of smooth functions on a manifold, by the convolution algebra of smooth functions on a Lie group, when such Lie group act on the manifold. In the case of a compact group, one recovers equivariant $K$-theory. One also obtains geometrically interesting results, for instance in the case of a discrete group.

LA - eng

KW - smooth circle actions on manifolds; cyclic homology; -algebra; convolution algebra; equivariant K-theory

UR - http://eudml.org/doc/74771

ER -

## References

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- [2] P. BAUM, J.-L. BRYLINSKI and R. MACPHERSON, Cohomologie équivariante délocalisée, C.R.A.S., t. 300, série I (1985), 605-608. Zbl0589.55003MR86g:55006
- [3] P. BAUM and A. CONNES, Geometric K-theory for Lie groups and foliations, Brown University-I.H.E.S., preprint, 1982. Zbl0985.46042
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- [8] A. CONNES, Cohomologie cyclique et foncteurs Extn, C.R.A.S., t. 296, série I (1983), 953-958. Zbl0534.18009MR86d:18007
- [9] B. L. FEIGIN and B.-L. TSYGAN, Cohomology of Lie algebras of generalized Jacobi matrices, Funct. Anal. and Appl., 17, n° 2 (1983), 86-87. Zbl0544.17011
- [10] P. JULG, K-théorie équivariante et produits croisés, C.R.A.S., t. 292, série I (1981), 629-632. Zbl0461.46044MR83b:46090
- [11] J.-L. KOSZUL, Sur certains groupes de transformations de Lie; Géométrie différentielle. Colloques Intern. du C.N.R.S., Strasbourg, 1953, 137-141. Zbl0101.16201MR15,600g
- [12] J.-L. LODAY and D. QUILLEN, Cyclic homology and the Lie algebra homology of matrices, Comment. Math. Helv., 59 (1984), 565-591. Zbl0565.17006MR86i:17003
- [13] G. SEGAL, Équivariant K-theory, Publ. Math. I.H.E.S., n° 34 (1968), 129-151. Zbl0199.26202MR38 #2769
- [14] D. BURGHELEA, The cyclic homology of the group rings, Comment. Math. Helv., 60 (1985), 354-365. Zbl0595.16022MR88e:18007

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