#### 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
##### Publication
Received: 24 March 2019
Revised: 11 September 2019
Accepted: 17 October 2019
Published: 30 September 2020
Proposed: András I Stipsicz
Seconded: Tomasz Mrowka, Ciprian Manolescu
##### Authors
 Peter Lambert-Cole School of Mathematics Georgia Institute of Technology Atlanta, GA United States http://www.math.gatech.edu/~plambertcole3