Volume 24, issue 3 (2020)

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Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture

Peter Lambert-Cole

Geometry & Topology 24 (2020) 1571–1614
Abstract

We develop new techniques for understanding surfaces in ${ℂℙ}^{2}$ 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 ${ℂℙ}^{2}$ 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.

Keywords
Thom conjecture, 4–manifolds, bridge trisections, minimal genus
Mathematical Subject Classification 2010
Primary: 57R17, 57R40