Bridge trisections in ℂℙ2 and the Thom conjecture

Lambert-Cole, Peter¹

¹ School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States

Geometry & topology, Tome 24 (2020) no. 3, pp. 1571-1614.

Résumé

We develop new techniques for understanding surfaces in ${ℂ\mathbb{P}}^2$ 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 ${ℂ\mathbb{P}}^2$ 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.

Détail

DOI : 10.2140/gt.2020.24.1571

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

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