Smith-Gysin Sequence

Abstract

Starting with a manifold $M$ and a semi-free action of $S^3$ on it, we have the Smith-Gysin sequence: $$\cdots \to H^{\ast }(M)\to H^{\ast -3}(M/S^3,M^{S^3})\oplus H^{\ast }(M^{S^3})\to H^{\ast +1}(M/S^3,M^{S^3})\to H^{\ast +1}(M)\to \cdots$$ 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 $M^{S^1}$: $$\cdots \to H^{\ast }(M)\to H^{\ast -3}(M/S^3,\Sigma /S^3)\oplus H^{\ast }(M^{S^3})\oplus {\left(H^{\ast -2}(M^{S^1})\right)}^{-{\mathbb{Z}}_2}\to H^{\ast +1}(M/S^3,M^{S^3})\to H^{\ast +1}(M)\to \cdots$$ Here, $\Sigma \subset M$ is the subset of points in $M$ whose isotropy groups are infinite. The group ${\mathbb{Z}}_2$ acts on $M^{S^1}$ by $j\in S^3$.

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