Intersection cohomology of stratified circle actions

Abstract

For any stratified pseudomanifold $X$ and any action of the unit circle $\mathbb{S}^1$ on $X$ preserving the stratification and the local structure, the orbit space $X/\mathbb{S}^1$ is also a stratified pseudomanifold. For each perversity $\overline{q}$ in $X$ the orbit map $\pi : X/\mathbb{S}^1$ induces a Gysin sequence relating the $\overline{q}$-intersection cohomologies of $X$ and $X/\mathbb{S}^1$. The third term of this sequence can be given by means of a spectral sequence on $X/\mathbb{S}^1 whose second term is the cohomology of the set of fixed points $X^{S^{1}}$ with values on a constructible sheaf.