Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Auckly, Dave Ruberman, Daniel |
| author_facet | Auckly, Dave Ruberman, Daniel |
| contents | We construct infinite rank summands isomorphic to $\mathbb{Z}^\infty$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with $S^2 \times S^2$. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give $\mathbb{Z}^\infty$ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups.
The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and $3$-manifolds in various $4$-manifolds, and for the space of positive scalar curvature metrics on standard PSC $4$-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory Auckly, Dave Ruberman, Daniel Geometric Topology Algebraic Topology Differential Geometry 57K41 (primary), 57R52, 53C21 (secondary) We construct infinite rank summands isomorphic to $\mathbb{Z}^\infty$ in the higher homotopy and homology groups of the diffeomorphism groups of certain $4$-manifolds. These spherical families become trivial in the homotopy and homology groups of the homeomorphism group; an infinite rank subgroup becomes trivial after a single stabilization by connected sum with $S^2 \times S^2$. The stabilization result gives rise to an inductive construction, starting from non-isotopic but pseudoisotopic diffeomorphisms constructed by the second author in 1998. The spherical families give $\mathbb{Z}^\infty$ summands in the homology of the classifying spaces of specific subgroups of those diffeomorphism groups. The non-triviality is shown by computations with family Seiberg-Witten invariants, including a gluing theorem adapted to our inductive construction. As applications, we we obtain infinite generation for higher homotopy and homology groups of spaces of embeddings of surfaces and $3$-manifolds in various $4$-manifolds, and for the space of positive scalar curvature metrics on standard PSC $4$-manifolds. |
| title | Families of diffeomorphisms, embeddings, and positive scalar curvature metrics via Seiberg-Witten theory |
| topic | Geometric Topology Algebraic Topology Differential Geometry 57K41 (primary), 57R52, 53C21 (secondary) |
| url | https://arxiv.org/abs/2501.11892 |