Smith-Gysin Sequence

Given a smooth semifree action of \(S^3\) on a manifold M, we have the Smith-Gysin sequence: \(\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} {\rightarrow } {H}^{^{*-3}}{\big ( M/S^3, M^{S^3} \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M/S^3, M^{S^3} \big )} {\rightarrow } {H}^{^{*+1}}{\big ( M \big )}\rightarrow \cdots \end{aligned}\) In this paper, we construct a Smith-Gysin sequence that does not require the semifree condition. This sequence includes a new term, referred to as the exotic term, which depends on the subset \(M^{S^1}\) : \(\begin{aligned} \cdots \rightarrow {H}^{^{*}}{\big ( M \big )} \rightarrow {H}^{^{*-3}}{\big ( M/S^3, \Sigma /S^3 \big )} \oplus {H}^{^{*}}{\big ( M^{S^3} \big )} \oplus \big ({H}^{^{*-2}}{\big ( M^{S^1} \big )}\big )^{-{\mathbb {Z}}_{_2}} \\ \rightarrow {H}^{^{*+1}}{\big ( M/S^3,M^{S^3} \big )} \rightarrow {H}^{^{*+1}}{\big ( M \big )} \rightarrow \cdots \end{aligned}\) Here, \(\Sigma \subset M\) is the subset of points in M whose isotropy group is infinite. The group \(\mathbb {Z}_2\) acts on \(M^{S^1}\) by \(j \in S^3\) .