Smith-Gysin Sequence
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866912220017852416 |
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| author | Prieto, J. I. Royo Aranguren, M. Saralegi Wolak, R. |
| author_facet | Prieto, J. I. Royo Aranguren, M. Saralegi Wolak, R. |
| contents | 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 $M^{S^1}$: $$ \cdots \to H^{*}(M) \to H^{*-3} (M/S^3, Σ/S^3) \oplus H^{*}(M^{S^3}) \oplus \left( H^{*-2}(M^{S^1})\right)^{-\mathbb{Z}_2} \to H^{*+1}(M/S^3,M^{S^3}) \to H^{*+1}(M) \to \cdots $$
Here, $Σ\subset M$ is the subset of points in $M$ whose isotropy groups are infinite. The group $\mathbb{Z}_2$ acts on $M^{S^1}$ by $j \in S^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_04309 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Smith-Gysin Sequence Prieto, J. I. Royo Aranguren, M. Saralegi Wolak, R. Algebraic Topology Differential Geometry 58D05 (Primary) 14F05, 55N10 (Secondary) 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 $M^{S^1}$: $$ \cdots \to H^{*}(M) \to H^{*-3} (M/S^3, Σ/S^3) \oplus H^{*}(M^{S^3}) \oplus \left( H^{*-2}(M^{S^1})\right)^{-\mathbb{Z}_2} \to H^{*+1}(M/S^3,M^{S^3}) \to H^{*+1}(M) \to \cdots $$ Here, $Σ\subset M$ is the subset of points in $M$ whose isotropy groups are infinite. The group $\mathbb{Z}_2$ acts on $M^{S^1}$ by $j \in S^3$. |
| title | Smith-Gysin Sequence |
| topic | Algebraic Topology Differential Geometry 58D05 (Primary) 14F05, 55N10 (Secondary) |
| url | https://arxiv.org/abs/2310.04309 |