The Gysin braid for $S^3$-actions on manifolds
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866918048492945408 |
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| author | Prieto, José Ignacio Royo Saralegi-Aranguren, Martintxo |
| author_facet | Prieto, José Ignacio Royo Saralegi-Aranguren, Martintxo |
| contents | Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic term depending on the subset $M^{\mathbb S^1}$.
Notice that the orbit space is a stratified pseudomanifold, a kind of singular spaces where intersection cohomology applies. In the case where the the action is semi-free, the first author has already constructed a Gysin sequence relating the cohomology of $M$ and the intersection cohomology of $M/\mathbb S^3$.
What happens if the action is not semi-free? This is the goal of this work.
The situation is more complicated and we do not find a Gysin sequence but a Gysin braid relating the cohomology of $M$ and the intersection cohomology of $M/\mathbb S^3$. This braid also contains an exotic term depending this time on the intersection cohomology of $M^{\mathbb S^1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_09002 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Gysin braid for $S^3$-actions on manifolds Prieto, José Ignacio Royo Saralegi-Aranguren, Martintxo Algebraic Topology 57R19, 57R30, 57S15 Given a smooth action of the sphere $\mathbb S^3$ on a manifold $M$, we have previously constructed a Gysin sequence relating the cohomology of the manifold $M$ and that of the orbit space $M/\mathbb S^3$. This sequence involves an exotic term depending on the subset $M^{\mathbb S^1}$. Notice that the orbit space is a stratified pseudomanifold, a kind of singular spaces where intersection cohomology applies. In the case where the the action is semi-free, the first author has already constructed a Gysin sequence relating the cohomology of $M$ and the intersection cohomology of $M/\mathbb S^3$. What happens if the action is not semi-free? This is the goal of this work. The situation is more complicated and we do not find a Gysin sequence but a Gysin braid relating the cohomology of $M$ and the intersection cohomology of $M/\mathbb S^3$. This braid also contains an exotic term depending this time on the intersection cohomology of $M^{\mathbb S^1}$. |
| title | The Gysin braid for $S^3$-actions on manifolds |
| topic | Algebraic Topology 57R19, 57R30, 57S15 |
| url | https://arxiv.org/abs/2301.09002 |