On localizing groups of exotic diffeomorphisms of 4-manifolds
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916358042681344 |
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| author | Konno, Hokuto Mallick, Abhishek |
| author_facet | Konno, Hokuto Mallick, Abhishek |
| contents | Ruberman in the 90's showed that the group of exotic diffeomorphisms of closed 4-manifolds can be infinitely generated. We provide various results on the question of when such infinite generation can localize to a smaller embedded submanifold of the original manifold. Our results include: (1) All known infinitely generated groups of exotic diffeomorphisms of 4-manifolds detected by families Seiberg-Witten theory do not localize to any topologically (locally-flatly) embedded rational homology balls in the ambient 4-manifold. (2) Many exotic diffeomorphisms cannot be obtained as Dehn twists along homology spheres (under mild assumptions). (3) There is no contractible 4-manifolds with Seifert fibered boundary that have a universal property for exotic diffeomorphisms analogous to a universal cork. In addition, there is no universal compact 4-manifold $W$ such that the set of exotic diffeomorphisms of a 4-manifold can localize to an embedding of $W$. (4) Certain infinite generations of exotic diffeomorphism groups do localize to a non-compact subset $V$ with a small Betti number, but not to any compact subset of $V$. (5) An analogous result holds for mapping class groups of 4-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11773 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On localizing groups of exotic diffeomorphisms of 4-manifolds Konno, Hokuto Mallick, Abhishek Geometric Topology Differential Geometry Ruberman in the 90's showed that the group of exotic diffeomorphisms of closed 4-manifolds can be infinitely generated. We provide various results on the question of when such infinite generation can localize to a smaller embedded submanifold of the original manifold. Our results include: (1) All known infinitely generated groups of exotic diffeomorphisms of 4-manifolds detected by families Seiberg-Witten theory do not localize to any topologically (locally-flatly) embedded rational homology balls in the ambient 4-manifold. (2) Many exotic diffeomorphisms cannot be obtained as Dehn twists along homology spheres (under mild assumptions). (3) There is no contractible 4-manifolds with Seifert fibered boundary that have a universal property for exotic diffeomorphisms analogous to a universal cork. In addition, there is no universal compact 4-manifold $W$ such that the set of exotic diffeomorphisms of a 4-manifold can localize to an embedding of $W$. (4) Certain infinite generations of exotic diffeomorphism groups do localize to a non-compact subset $V$ with a small Betti number, but not to any compact subset of $V$. (5) An analogous result holds for mapping class groups of 4-manifolds. |
| title | On localizing groups of exotic diffeomorphisms of 4-manifolds |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2406.11773 |