Intersection cohomology of 𝑆¹-actions

Abstract

Given a free action of the circle S 1 {{\mathbf {S}}^1} on a differentiable manifold M M , there exists a long exact sequence that relates the cohomology of M M with the cohomology of the manifold M / S 1 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 / S 1 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 / S 1 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 ] [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.