Equivariant Morse Homology for Reflection Actions via Broken Trajectories
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913068232998912 |
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| author | Bao, Erkao Lawson, Tyler Liu, Lina |
| author_facet | Bao, Erkao Lawson, Tyler Liu, Lina |
| contents | We consider a finite group $G$ acting on a manifold $M$. For any equivariant Morse function, which is a generic condition, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse-Smale. Here, the pair $(f,g)$ is called Morse-Smale if the descending and ascending manifolds intersect transversely. The best possible metrics $g$ are those that make the pair $(f,g)$ stably Morse-Smale.
A diffeomorphism $ϕ: M \to M$ is a reflection, if $ϕ^2 = \operatorname{id}$ and the fixed point set of $ϕ$ forms a codimension-one submanifold (with $M \setminus M^{\operatorname{fix}}$ not necessarily disconnected).
In this note, we focus on the special case where the group $G = \{\operatorname{id}, ϕ\}$. We show that the condition of being stably Morse-Smale is generic for metrics $g$. Given a stably Morse-Smale pair, we introduce a canonical equivariant Thom-Smale-Witten complex by counting certain broken trajectories.
This has applications to the case when we have a manifold with boundary and when the Morse function has critical points on the boundary.
We provide an alternative definition of the Thom-Smale-Witten complexes, which are quasi-isomorphic to those defined by Kronheimer and Mrowka.
We also explore the case when $G$ is generated by multiple reflections. As an example, we compute the Thom-Smale-Witten complex of an upright higher-genus surface by counting broken trajectories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_16924 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Equivariant Morse Homology for Reflection Actions via Broken Trajectories Bao, Erkao Lawson, Tyler Liu, Lina Geometric Topology Symplectic Geometry Primary 55N25, Secondary 55M35, 55N91, 53D40 We consider a finite group $G$ acting on a manifold $M$. For any equivariant Morse function, which is a generic condition, there does not always exist an equivariant metric $g$ on $M$ such that the pair $(f,g)$ is Morse-Smale. Here, the pair $(f,g)$ is called Morse-Smale if the descending and ascending manifolds intersect transversely. The best possible metrics $g$ are those that make the pair $(f,g)$ stably Morse-Smale. A diffeomorphism $ϕ: M \to M$ is a reflection, if $ϕ^2 = \operatorname{id}$ and the fixed point set of $ϕ$ forms a codimension-one submanifold (with $M \setminus M^{\operatorname{fix}}$ not necessarily disconnected). In this note, we focus on the special case where the group $G = \{\operatorname{id}, ϕ\}$. We show that the condition of being stably Morse-Smale is generic for metrics $g$. Given a stably Morse-Smale pair, we introduce a canonical equivariant Thom-Smale-Witten complex by counting certain broken trajectories. This has applications to the case when we have a manifold with boundary and when the Morse function has critical points on the boundary. We provide an alternative definition of the Thom-Smale-Witten complexes, which are quasi-isomorphic to those defined by Kronheimer and Mrowka. We also explore the case when $G$ is generated by multiple reflections. As an example, we compute the Thom-Smale-Witten complex of an upright higher-genus surface by counting broken trajectories. |
| title | Equivariant Morse Homology for Reflection Actions via Broken Trajectories |
| topic | Geometric Topology Symplectic Geometry Primary 55N25, Secondary 55M35, 55N91, 53D40 |
| url | https://arxiv.org/abs/2411.16924 |