Bridge trisections in $\mathbb{CP}^2$ and the Thom conjecture (with Corrigendum)

TL;DR

This paper introduces bridge trisection techniques for surfaces in rac{2}{2}CP^2 and provides a new, gauge-theory-free proof of the Thom conjecture, with a corrigendum addressing a critical error in the main theorem.

Contribution

It develops new methods using bridge trisections to prove the Thom conjecture without gauge theory or pseudoholomorphic curves.

Findings

New proof of the Thom conjecture avoiding gauge theory.

Development of bridge trisection techniques for surfaces in rac{2}{2}CP^2.

Identification of a fatal error in the main theorem's proof, with remaining results still valid.

Abstract

In this paper, we develop new techniques for understanding surfaces in $\mathbb{CP}^2$ 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 $\mathbb{CP}^2$ 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. Corrigendum: This paper contains a fatal error in the proof of Theorem 1.1, which is the headline result of the paper. The error is localized to Section 6 and is described in a Corrigendum at the end of this updated version. The remaining results in Sections 1 through 5 remain valid.