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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Intersection cohomology of $ S\sp 1$-actions

Authors: Gilbert Hector and Martin Saralegi
Journal: Trans. Amer. Math. Soc. 338 (1993), 263-288
MSC: Primary 57S15; Secondary 55N33, 57N80
MathSciNet review: 1116314
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Abstract: Given a free action of the circle $ {{\mathbf{S}}^1}$ on a differentiable manifold $ M$, there exists a long exact sequence that relates the cohomology of $ M$ with the cohomology of the manifold $ M/{{\mathbf{S}}^1}$. This is the Gysin sequence. This result is still valid if we allow the action to have stationary points.

In this paper we are concerned with actions where fixed points are allowed. Here the quotient space $ M/{{\mathbf{S}}^1}$ is no longer a manifold but a stratified pseudomanifold (in terms of Goresky and MacPherson). We get a similar Gysin sequence where the cohomology of $ M/{{\mathbf{S}}^1}$ is replaced by its intersection cohomology. As in the free case, the connecting homomorphism is given by the product with the Euler class $ [e]$. Also, the vanishing of this class is related to the triviality of the action. In this Gysin sequence we observe the phenomenon of perversity shifting. This is due to the allowability degree of the Euler form.

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  • [1] J. P. Brasselet, G. Hector, and M. Saralegi, Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom. 9 (1991), 211-243. MR 1143404 (93g:55009)
  • [2] G. Bredon, Introduction to compact transformation groups, Pure and Appl. Math., Academic Press, New York and London, 1972. MR 0413144 (54:1265)
  • [3] J. L. Brylinsky, Equivariant intersection cohomology, Inst. Hautes Études Sci. preprint., June 1986.
  • [4] M. Goresky and R. MacPherson, Intersection homology theory, Topology 19 (1980), 135-162. MR 572580 (82b:57010)
  • [5] -, Intersection homology. II, Invent. Math. 71 (1983), 77-129. MR 696691 (84i:57012)
  • [6] W. Greub, S. Halperin, and R. Vanstone, Connections, curvature, and cohomology, Pure and Appl. Math., Academic Press, New York and London, 1972. MR 0400275 (53:4110)
  • [7] M. Nicolau and A. Reventós, On some geometrical properties of Seifert bundles, Israel J. Math. 47 (1984), 323-334. MR 764302 (87a:53068)
  • [8] M. Saralegi, The Euler class for flows of isometries, Research Notes in Math. 131 (1985), 220-227. MR 864872 (88a:58163)
  • [9] -, Homological properties of unfoldable stratified pseudomanifolds, Illinois J. Math. (to appear).
  • [10] M. Saralegi and G. Hector, Formes différentielles d'intersection des prestratifications abstraites, C. R. Acad. Sci. Paris 308 (1989), 25-28. MR 980316 (90a:58007)
  • [11] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171-188. MR 0321133 (47:9666)
  • [12] D. Tischler, On fibering certain foliated manifolds over $ {{\mathbf{S}}^1}$, Topology 9 (1980), 153-154. MR 0256413 (41:1069)
  • [13] A. Verona, Stratified mappings--Structure and triangulability, Lecture Notes in Math., vol. 1102, Springer-Verlag, 1984. MR 771120 (86k:58010)

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Keywords: Gysin sequence, Euler class, stratified spaces
Article copyright: © Copyright 1993 American Mathematical Society

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