On integral rigidity in Seiberg-Witten theory
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866914088299266048 |
|---|---|
| author | Lin, Francesco Eismeier, Mike Miller |
| author_facet | Lin, Francesco Eismeier, Mike Miller |
| contents | We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed $4$-manifold $X$ containing a non-separating hypersurface $Y$ satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if $X$ has the homology of a four-torus, and it contains a non-separating three-torus, then the sum of all Seiberg-Witten invariants of $X$ is determined in purely cohomological terms. Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson's TQFT approach to the formula of Meng-Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg-Witten equations on $X$ and reducible ones on $Y$ and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline{HM}_*$ induced by a negative cobordism between three-manifolds, which might be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_17884 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On integral rigidity in Seiberg-Witten theory Lin, Francesco Eismeier, Mike Miller Geometric Topology We introduce a framework to prove integral rigidity results for the Seiberg-Witten invariants of a closed $4$-manifold $X$ containing a non-separating hypersurface $Y$ satisfying suitable (chain-level) Floer theoretic conditions. As a concrete application, we show that if $X$ has the homology of a four-torus, and it contains a non-separating three-torus, then the sum of all Seiberg-Witten invariants of $X$ is determined in purely cohomological terms. Our results can be interpreted as $(3+1)$-dimensional versions of Donaldson's TQFT approach to the formula of Meng-Taubes, and build upon a subtle interplay between irreducible solutions to the Seiberg-Witten equations on $X$ and reducible ones on $Y$ and its complement. Along the way, we provide a concrete description of the associated graded map (for a suitable filtration) of the map on $\overline{HM}_*$ induced by a negative cobordism between three-manifolds, which might be of independent interest. |
| title | On integral rigidity in Seiberg-Witten theory |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2409.17884 |