The Gysin Sequence for $S^1$-actions on stratified pseudomanifolds

Abstract

For any stratified pseudomanifold $X$ and any suitable action of the unit circle $S^1$ on $X$ preserving the strata and the local topological structure, the orbit space $B=X/S^1$ is again a stratified pseudomanifold and the orbit map $π/X\to B$ is a stratified morphism. For each perversity in $X$ this arrow induces a long exact sequence relating the intersection cohomologies of $X$ and $B$; we call it the Gysin sequence of $X$ induced by the action. The third term of the Gysin sequence is called the Gysin term; its cohomology depends on basic global and local data. Global data concerns the intersection cohomology of $B$ and the Euler class induced by the action; while local data is related to the Euler class of the links of the fixed strata. The cohomology of the Gysin term is calculated trhough a residual constructible sheaf in $B$ with support in the fixed points set. In the last modification of this paper, we improved the definition of the Euler class in intersection cohomology, which is made by recursion and implies the definition of an Euler perversity. We also verify that the Euler class is functorial for a suitable family of equivariant morphisms (i.e., not every equivariant morphism preserves the Euler class).