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Bridge trisections in \(\mathbb{CP}^2\) and the Thom conjecture. (English) Zbl 07256611

Summary: 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.

MSC:

57K40 General topology of 4-manifolds
57R17 Symplectic and contact topology in high or arbitrary dimension
57R40 Embeddings in differential topology
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