Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture (with Corrigendum)
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866913725776134144 |
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| author | Lambert-Cole, Peter |
| author_facet | Lambert-Cole, Peter |
| contents | In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $\mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques.
Corrigendum: This paper contains a fatal error in the proof of Theorem 1.1, which is the headline result of the paper. The error is localized to Section 6 and is described in a Corrigendum at the end of this updated version. The remaining results in Sections 1 through 5 remain valid. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1807_10131 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture (with Corrigendum) Lambert-Cole, Peter Geometric Topology 57R17, 57R40 In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ via bridge trisections. Trisections are a novel approach to smooth 4-manifold topology, introduced by Gay and Kirby, that provide an avenue to apply 3-dimensional tools to 4-dimensional problems. Meier and Zupan subsequently developed the theory of bridge trisections for smoothly embedded surfaces in 4-manifolds. The main application of these techniques is a new proof of the Thom conjecture, which posits that algebraic curves in $\mathbb{CP}^2$ have minimal genus among all smoothly embedded, oriented surfaces in their homology class. This new proof is notable as it completely avoids any gauge theory or pseudoholomorphic curve techniques. Corrigendum: This paper contains a fatal error in the proof of Theorem 1.1, which is the headline result of the paper. The error is localized to Section 6 and is described in a Corrigendum at the end of this updated version. The remaining results in Sections 1 through 5 remain valid. |
| title | Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture (with Corrigendum) |
| topic | Geometric Topology 57R17, 57R40 |
| url | https://arxiv.org/abs/1807.10131 |