Intersection Cohomology of S1-Actions on Pseudomanifolds

Abstract

For any smooth free action of the unit circle S1 on a smooth manifold M, the Gysin sequence of M is a long exact sequence relating the DeRham Cohomology of M and the orbit space M/S1. If the action is not free then M/S1 is not a smooth manifold but a stratified pseudomanifold, the lenght of M/S1 depending on the number of orbit types; and there is a Gysin sequence relating their intersection cohomologies. The links of the fixed strata in M/S1 are cohomological complex projective spaces, so the conecting homomorphism of this sequences is the multiplication by the Euler class. In this article we extend the above results for any action of S1 on a stratified pseudomanifold X of lenght 1. We use the DeRham-like intersection cohomology defined by means of an unfolding. If the action preserves the local structure, then the orbit space X/S1 is again a stratified pseudomanifold of lenght 1 and has an unfolding. There is a long exact sequence relating the intersection cohomology of X and X/S1 with a third complex $\mathcal{G}$, the Gysin Term, whose cohomology depends on basic cohomological data of two flavours: global and local. Global data concerns the Euler class induced by the action; local information depends on the cohomology of the fixed strata with values on some presheaves.