Which finite groups act smoothly on a given 4-manifold?
We prove that for any closed smooth 4-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G<\operatorname{Diff}(X)$ has a subgroup $N$ which is abelian or nilpotent of class 2 , and which satisfies $[G: N] \leq C$. We give sufficient conditions on $X$ for $\opera...
| Autores: | , |
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| Tipo de recurso: | artículo |
| Estado: | Versión aceptada para publicación |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universidad de Barcelona |
| Repositorio: | Dipòsit Digital de la UB |
| OAI Identifier: | oai:diposit.ub.edu:2445/195522 |
| Acceso en línea: | https://hdl.handle.net/2445/195522 |
| Access Level: | acceso abierto |
| Palabra clave: | Transformacions (Matemàtica) Varietats (Matemàtica) Topologia de baixa dimensió Varietats simplèctiques Transformations (Mathematics) Manifolds (Mathematics) Low-dimensional topology Symplectic manifolds |
| Sumario: | We prove that for any closed smooth 4-manifold $X$ there exists a constant $C$ with the property that each finite subgroup $G<\operatorname{Diff}(X)$ has a subgroup $N$ which is abelian or nilpotent of class 2 , and which satisfies $[G: N] \leq C$. We give sufficient conditions on $X$ for $\operatorname{Diff}(X)$ to be Jordan, meaning that there exists a constant $C$ such that any finite subgroup $G<\operatorname{Diff}(X)$ has an abelian subgroup $A$ satisfying $[G: A] \leq C$. Some of these conditions are homotopical, such as having nonzero Euler characteristic or nonzero signature, others are geometric, such as the absence of embedded tori of arbitrarily large self-intersection arising as fixed point components of periodic diffeomorphisms. Relying on these results, we prove that: (1) the symplectomorphism group of any closed symplectic 4-manifold is Jordan, and (2) the automorphism group of any almost complex closed 4-manifold is Jordan. |
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