Bridge trisections in ℂℙ2 and the Thom conjecture

Lambert-Cole, Peter

doi:10.2140/gt.2020.24.1571

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

Peter Lambert-Cole

Geometry & Topology 24 (2020) 1571–1614

DOI: 10.2140/gt.2020.24.1571

Abstract

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

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Keywords

Thom conjecture, 4–manifolds, bridge trisections, minimal genus

Mathematical Subject Classification 2010

Primary: 57R17, 57R40

References

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

Volume 24, issue 3 (2020)