Smith-Gysin sequence (Q6616256)

Let \(M\) be a smooth manifold without boundary. Assume the group \(S^3\) acts smoothly on \(M\). If the action is free, there is a long exact sequence \N\[\N\cdots \to H^\ast(M)\to H^{\ast-3}(M/S^3)\to H^{\ast+1}(M/S^3)\to H^{\ast +1}(M)\to\cdots \N\]\Nrelating the cohomology classes of \(M\) to those of the quotient space \(M/S^3\). The sequence is known as the Gysin sequence. If the action is semifree, i.e., if it is free outside the fixed point set \(M^{S^3}\), there is a long exact sequence\N\begin{align*}\N\cdots &\to H^\ast(M)\to H^{\ast -3}(M/S^3, M^{S^3})\oplus H^\ast(M^{S^3})\\\N&\to H^{\ast +1}(M/S^3, M^{S^3}) \to H^{\ast +1}(M)\to \cdots\N\end{align*}\Nknown as the Smith-Gysin sequence.\N\NIn this paper, the authors expand the scope of the Smith-Gysin sequence to any smooth, effective action of \(S^3\) on \(M\). Denote the fixed point set of the subgroup \(S^1\) of \(S^3\) by \(M^{S^1}\). The group \({\mathbb{Z}}_2={\mathbb{Z}}/2{\mathbb{Z}}\) acts on \(M^{S^1}\) by the quaternion \(j\in S^3\). The authors obtain the following long exact sequence associated to the \(S^3\)-action:\N\begin{align*}\N\cdots & \to H^\ast(M)\to H^{\ast-3}( M/S^3, \Sigma/S^3)\oplus \Bigl( H^{\ast-2}(M^{S^1})\Bigr)^{-{\mathbb{Z}}_2} \oplus H^\ast(M^{S^3})\\\N&\to H^{\ast +1}(M/{S^3}, M^{S^3})\to H^{\ast +1}(M)\to\cdots\N\end{align*}\NHere \(\Sigma=\{ x\in M\mid \dim S^3_x >0\}\) and \((-)^{-{\mathbb{Z}}_2}\) denotes the subspace of antisymmetric elements. The authors also prove that there is a long exact sequence\N\begin{align*}\N\cdots & \to H^\ast(M/S^3, M^{S^3})\to H^\ast(M/S^3)\oplus H^\ast (M, M^{S^3})\\\N&\to H^\ast(M) \to H^{\ast+1}(M/S^3,M^{S^3})\to \cdots\N\end{align*}\NThe cohomology \(H^\ast(-)\) refers to the singular cohomology with real coefficients.\N\NFor the entire collection see [Zbl 1532.53004].