  For any stratified pseudomanifold $X$ and any suitable action of the unit
circle $S^1$ on $X$ preserving the strata and the local topological structure,
the orbit space $B=X/S^1$ is again a stratified pseudomanifold and the orbit
map $\pi/X\to B$ is a stratified morphism. For each perversity in $X$ this
arrow induces a long exact sequence relating the intersection cohomologies of
$X$ and $B$; we call it the Gysin sequence of $X$ induced by the action.
  The third term of the Gysin sequence is called the Gysin term; its cohomology
depends on basic global and local data. Global data concerns the intersection
cohomology of $B$ and the Euler class induced by the action; while local data
is related to the Euler class of the links of the fixed strata. The cohomology
of the Gysin term is calculated trhough a residual constructible sheaf in $B$
with support in the fixed points set.
  In the last modification of this paper, we improved the definition of the
Euler class in intersection cohomology, which is made by recursion and implies
the definition of an Euler perversity. We also verify that the Euler class is
functorial for a suitable family of equivariant morphisms (i.e., not every
equivariant morphism preserves the Euler class).
