  For any smooth free action of the unit circle S1 on a smooth manifold M, the
Gysin sequence of M is a long exact sequence relating the DeRham Cohomology of
M and the orbit space M/S1. If the action is not free then M/S1 is not a smooth
manifold but a stratified pseudomanifold, the lenght of M/S1 depending on the
number of orbit types; and there is a Gysin sequence relating their
intersection cohomologies. The links of the fixed strata in M/S1 are
cohomological complex projective spaces, so the conecting homomorphism of this
sequences is the multiplication by the Euler class.
  In this article we extend the above results for any action of S1 on a
stratified pseudomanifold X of lenght 1. We use the DeRham-like intersection
cohomology defined by means of an unfolding. If the action preserves the local
structure, then the orbit space X/S1 is again a stratified pseudomanifold of
lenght 1 and has an unfolding. There is a long exact sequence relating the
intersection cohomology of X and X/S1 with a third complex $\mathcal{G}$, the
Gysin Term, whose cohomology depends on basic cohomological data of two
flavours: global and local. Global data concerns the Euler class induced by the
action; local information depends on the cohomology of the fixed strata with
values on some presheaves.
