The Gysin braid for S3-actions onmanifolds

In a previous work, we constructed a Gysin sequence that relates the cohomology of a manifold M to that of the orbit space M/S 3 , where the sphere S 3 acts smoothly on M.This sequence includes an exotic term that depends on M S 1 , the subset of points fixed by the action of the subgroup S 1 .The orbit space is a stratified pseudomanifold, which is a type of singular space where intersection cohomology can be applied.When the action is semifree, Saralegi-Aranguren has already constructed a Gysin sequence that relates the cohomology of M to the intersection cohomology of M/S 3 .However, what happens when the action is not semifree?This is the main focus of this work.The situation becomes more complex, and we do not find just a Gysin sequence.Instead, we construct a Gysin braid that relates the cohomology of M to the intersection cohomology of M/S 3 .This braid also contains an exotic term that depends on the intersection cohomology of the fixed point subset M S 1 .1. Intersection cohomology 301 2. Invariant differential forms 307 3. The integration operator / 317 4. Gysin braid for a mobile action 327 5. Gysin sequence for a nonmobile action 335