The Gysin braid for $S^{3}$-actions on manifolds

José Ignacio Royo Prieto and Martintxo Saralegi-Aranguren

Vol. 337 (2025), No. 2, 297–337

DOI: 10.2140/pjm.2025.337.297

Abstract

In a previous work, we constructed a Gysin sequence that relates the cohomology of a manifold $M$ to that of the orbit space $M∕S^3$, where the sphere $S^3$ acts smoothly on $M$. This sequence includes an exotic term that depends on $M^{S^1}$, the subset of points fixed by the action of the subgroup $S^1$.

The orbit space is a stratified pseudomanifold, which is a type of singular space where intersection cohomology can be applied. When the action is semifree, Saralegi-Aranguren has already constructed a Gysin sequence that relates the cohomology of $M$ to the intersection cohomology of $M∕S^3$.

However, what happens when the action is not semifree? This is the main focus of this work. The situation becomes more complex, and we do not find just a Gysin sequence. Instead, we construct a Gysin braid that relates the cohomology of $M$ to the intersection cohomology of $M∕S^3$. This braid also contains an exotic term that depends on the intersection cohomology of the fixed point subset $M^{S^1}$.

Keywords

Gysin sequence, intersection cohomology, $S^{3}$-actions

Mathematical Subject Classification

Primary: 57S15

Secondary: 55N33

Milestones

Received: 22 October 2024

Revised: 6 April 2025

Accepted: 3 May 2025

Published: 3 June 2025