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.