  Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have
previously constructed a Gysin sequence relating the cohomology of the manifold
$M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an
exotic term depending on the subset $M^{\mathbb S^1}$.
  Notice that the orbit space is a stratified pseudomanifold, a kind of
singular spaces where intersection cohomology applies. In the case where the
the action is semi-free, the first author has already constructed a Gysin
sequence relating the cohomology of $M$ and the intersection cohomology of
$M/\mathbb S^3$.
  What happens if the action is not semi-free? This is the goal of this work.
  The situation is more complicated and we do not find a Gysin sequence but a
Gysin braid relating the cohomology of $M$ and the intersection cohomology of
$M/\mathbb S^3$. This braid also contains an exotic term depending this time on
the intersection cohomology of $M^{\mathbb S^1}$.
