Bridge trisections in ${\mathbb{CP}}^2$ and the Thom conjecture (with Corrigendum)

Abstract

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.

33 pages, 18 figures

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• Trisecting 4-manifolds
• Bridge trisections of knotted surfaces in $S^4$
• Bridge trisections of knotted surfaces in 4--manifolds
• Genus two trisections are standard
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Cited by in corpus (4)

• An Algorithm taking Kirby diagrams to Trisection diagrams
• Symplectic 4-manifolds admit Weinstein trisections
• The Minimal Genus Problem
• Bounding the Kirby-Thompson invariant of spun knots