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.

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)