Smith-Gysin Sequence

TL;DR

This paper extends the Smith-Gysin sequence to non-semi-free $S^3$ actions on manifolds, introducing an exotic term involving the $S^1$ fixed points and their $bZ_2$-action.

Contribution

It constructs a generalized Smith-Gysin sequence that includes a new term accounting for points with infinite isotropy, removing the semi-free restriction.

Findings

Introduces a new 'exotic' term in the sequence involving $H^{*-2}(M^{S^1})^{-bZ_2}$.

Provides a sequence applicable to more general $S^3$ actions beyond semi-free cases.

Enhances understanding of the topology of manifolds with $S^3$ symmetry.

Abstract

Starting with a manifold $M$ and a semi-free action of $S^3$ on it, we have the Smith-Gysin sequence: $$ \cdots \to H^{*}( M) \to H^{*-3}(M/S^3, M^{S^3}) \oplus H^{*} (M^{S^3}) \to H^{*+1}(M/S^3, M^{S^3}) \to H^{*+1}(M) \to \cdots $$ 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 $M^{S^1}$: $$ \cdots \to H^{*}(M) \to H^{*-3} (M/S^3, \Sigma/S^3) \oplus H^{*}(M^{S^3}) \oplus \left( H^{*-2}(M^{S^1})\right)^{-\mathbb{Z}_2} \to H^{*+1}(M/S^3,M^{S^3}) \to H^{*+1}(M) \to \cdots $$ 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 $j \in S^3$.