Constructing locally flat surfaces in 4-manifolds
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911305597714432 |
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| author | Ray, Arunima |
| author_facet | Ray, Arunima |
| contents | There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's disc embedding theorem. In this expository article, we give an introduction to these methods by sketching proofs of the following results: every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus (Lee-Wilczynski); and every Alexander polynomial one knot in $S^3$ is topologically slice (Freedman-Quinn). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_18423 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constructing locally flat surfaces in 4-manifolds Ray, Arunima Geometric Topology 57K40, 57N35 There are two main approaches to building locally flat embedded surfaces in 4-manifolds: direct methods which geometrically manipulate a given map of a surface, and more indirect methods using surgery theory. Both rely on Freedman-Quinn's disc embedding theorem. In this expository article, we give an introduction to these methods by sketching proofs of the following results: every primitive second homology class in a closed, simply connected 4-manifold is represented by a locally flat embedded torus (Lee-Wilczynski); and every Alexander polynomial one knot in $S^3$ is topologically slice (Freedman-Quinn). |
| title | Constructing locally flat surfaces in 4-manifolds |
| topic | Geometric Topology 57K40, 57N35 |
| url | https://arxiv.org/abs/2412.18423 |