Intersection Cohomology of S1-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 lenght 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 flavours: 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. Foreword A pseudomanifold is a topological space X with two features. First, there is a closed Σ ⊂ X called the singular part, which is the disjoint union of smooth manifolds. The X −Σ is a dense smooth manifold. We call strata the connected components of Σ and X − Σ; they constitute a locally finite partition of X. The second feature is the local conical behavior of X, the model being a product U × c(L) of a smooth manifold U with the open cone of a compact smooth manifold L called the link of U . A careful reader will notice that stratified pseudomanifolds with arbitrary lenght have a richer and more complicated topological structure; in this article we deal with stratified pseudomanifolds of lenght ≤ 1, which we call just pseudomanifolds. Between the various ways for defining the intersection cohomology; the reader can see [4] for a definition in pl-stratified pseudomanifolds; [5], [9], for a definition with sheaves; [11] for an approach with L-cohomology; [3] for an exposition in Thom-Mather spaces. In this article, we use the DeRham-like definition exposed in [18]. We work with differential forms in X − Σ and measure their behavior when approaching to Σ, trough an auxiliary construction called an unfolding of X. Although X may have many different unfoldings, its intersection cohomology does not depend on any particular choice. When S1 acts on X preserving the local structure then the orbit space X/S1 is again a pseudomanifold with an unfolding. Date: May 9/2002. 1991 Mathematics Subject Classification. 35S35; 55N33.