Cyclic homology and equivariant theories

Jean-Luc Brylinski

Annales de l'institut Fourier (1987)

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

Abstract

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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.

How to cite

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Brylinski, 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. [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
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  7. [7] A. CONNES, Non-commutative differential geometry, Publ. Math. I.H.E.S., 62 (1986), 257-360. 
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  10. [10] P. JULG, K-théorie équivariante et produits croisés, C.R.A.S., t. 292, série I (1981), 629-632. Zbl0461.46044MR83b:46090
  11. [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. [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. [13] G. SEGAL, Équivariant K-theory, Publ. Math. I.H.E.S., n° 34 (1968), 129-151. Zbl0199.26202MR38 #2769
  14. [14] D. BURGHELEA, The cyclic homology of the group rings, Comment. Math. Helv., 60 (1985), 354-365. Zbl0595.16022MR88e:18007

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