Smith-Gysin Sequence

Starting with a manifold $M$ and a semi-free action of $S^3$ on it, we have the Smith-Gysin sequence: $$\cdots \to H^{\ast }(M)\to H^{\ast -3}(M\mathrm{/}{S}^3,M^{S^3})\oplus H^{\ast }(M^{S^3})\to H^{\ast +1}(M\mathrm{/}{S}^3,M^{S^3})\to H^{\ast +1}(M)\to \cdots$$ In this paper, we construct a Smith-Gysin sequence that does not require the semi-free condition. This sequence includes a new term, referred to as the “exotic term,” which depends on the subset $M^{S^1}$: $$\cdots \to H^{\ast }(M)\to H^{\ast -3}(M\mathrm{/}{S}^3,\mathrm{\Sigma }\mathrm{/}{S}^3)\oplus H^{\ast }(M^{S^3})\oplus {(H^{\ast -2}(M^{S^1}))}^{-{\mathbb{Z}}_2}\to H^{\ast +1}(M\mathrm{/}{S}^3,M^{S^3})\to H^{\ast +1}(M)\to \cdots$$ Here, $\mathrm{\Sigma }\subset M$ is the subset of points in $M$ whose isotropy groups are infinite. The group ${\mathbb{Z}}_2$ acts on $M^{S^1}$ by $j\in S^3$.