Intersection cohomology of 𝑆¹-actions

Hector, Gilbert; Saralegi, Martin

doi:10.1090/S0002-9947-1993-1116314-7

Intersection cohomology of $S^ 1$-actions
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Trans. Amer. Math. Soc. 338 (1993), 263-288 Request permission

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.

References

J.-P. Brasselet, G. Hector, and M. Saralegi, Théorème de de Rham pour les variétés stratifiées, Ann. Global Anal. Geom. 9 (1991), no. 3, 211–243 (French). MR 1143404, DOI 10.1007/BF00136813
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144

Equivariant intersection cohomology

Mark Goresky and Robert MacPherson, Intersection homology theory, Topology 19 (1980), no. 2, 135–162. MR 572580, DOI 10.1016/0040-9383(80)90003-8
Mark Goresky and Robert MacPherson, Intersection homology. II, Invent. Math. 72 (1983), no. 1, 77–129. MR 696691, DOI 10.1007/BF01389130
Werner Greub, Stephen Halperin, and Ray Vanstone, Connections, curvature, and cohomology, Pure and Applied Mathematics, Vol. 47-III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Volume III: Cohomology of principal bundles and homogeneous spaces. MR 0400275
M. Nicolau and A. Reventós, On some geometrical properties of Seifert bundles, Israel J. Math. 47 (1984), no. 4, 323–334. MR 764302, DOI 10.1007/BF02760606
M. Saralegui, The Euler class for flows of isometries, Differential geometry (Santiago de Compostela, 1984) Res. Notes in Math., vol. 131, Pitman, Boston, MA, 1985, pp. 220–227. MR 864872

Homological properties of unfoldable stratified pseudomanifolds

Martin Saralegi and Gilbert Hector, Formes différentielles d’intersection: un théorème de de Rham pour l’homologie d’intersection des préstratifications abstraites, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 1, 25–28 (French, with English summary). MR 980316
Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. MR 321133, DOI 10.1090/S0002-9947-1973-0321133-2
D. Tischler, On fibering certain foliated manifolds over $S^{1}$, Topology 9 (1970), 153–154. MR 256413, DOI 10.1016/0040-9383(70)90037-6
Andrei Verona, Stratified mappings—structure and triangulability, Lecture Notes in Mathematics, vol. 1102, Springer-Verlag, Berlin, 1984. MR 771120, DOI 10.1007/BFb0101672