-actions on Pseudomanifolds

For any smooth free action of the unit circle S in a manifold M ; the Gysin sequence of M is a long exact sequence relating the DeRham cohomologies of M and its orbit space M/S. If the action is not free then M/S is not a manifold but a stratified pseudomanifold and there is a Gysin sequence relating the DeRham cohomology of M with the intersection cohomology of M/S. In this work we extend the above statements for any stratified pseudomanifold X of length 1, whenever the action of S preserves the local structure. We give a Gysin sequence relating the intersection cohomologies of X and X/S with a third term G, the Gysin term; whose cohomology depends on basic cohomological data of two flavors: global data concerns the Euler class induced by the action, local data relates the Gysin term and the cohomology of the fixed strata with values on a locally trivial presheaf.