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

Peter Lambert-Cole

TL;DR

The paper develops bridge trisections for knotted surfaces in CP^2, leveraging a genus-1 CP^2 trisection compatible with toric geometry to translate 4D questions into 3D contact-geometric data. By introducing algebraic and geometric notions of transversality and constructing torus diagrams, it reduces the Thom conjecture to a ribbon-Bennequin-type inequality within a purely 3D framework, and shows how adjunction-type bounds follow from diagrammatic invariants. A sequence of stabilization and braiding moves, plus a detailed treatment of simple clasps, enables an isotopy-from-algebraic-transversality to a geometrically transverse setting, yielding the genus bound g(𝒦) ≥ 1/2(d−1)(d−2). However, a Corrigendum notes a fatal error localized to Section 6, affecting Theorem 1.1 (and related results) while leaving Sections 1–5 intact; the results pertaining to the Thom conjecture via the local-to-global strategy remain true in the literature, with the core 3D argument motivating alternative proofs. The work thus highlights a promising 3D-contact-geometry route to adjunction-type results in 4-manifolds, and suggests further refinement of the transverse-bride framework and its topological implications.

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.

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

TL;DR

The paper develops bridge trisections for knotted surfaces in CP^2, leveraging a genus-1 CP^2 trisection compatible with toric geometry to translate 4D questions into 3D contact-geometric data. By introducing algebraic and geometric notions of transversality and constructing torus diagrams, it reduces the Thom conjecture to a ribbon-Bennequin-type inequality within a purely 3D framework, and shows how adjunction-type bounds follow from diagrammatic invariants. A sequence of stabilization and braiding moves, plus a detailed treatment of simple clasps, enables an isotopy-from-algebraic-transversality to a geometrically transverse setting, yielding the genus bound g(𝒦) ≥ 1/2(d−1)(d−2). However, a Corrigendum notes a fatal error localized to Section 6, affecting Theorem 1.1 (and related results) while leaving Sections 1–5 intact; the results pertaining to the Thom conjecture via the local-to-global strategy remain true in the literature, with the core 3D argument motivating alternative proofs. The work thus highlights a promising 3D-contact-geometry route to adjunction-type results in 4-manifolds, and suggests further refinement of the transverse-bride framework and its topological implications.

Abstract

In this paper, we develop new techniques for understanding surfaces in 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 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.

Paper Structure

This paper contains 30 sections, 35 theorems, 82 equations, 18 figures.

Table of Contents

  1. Introduction
  2. Trisections
  3. Trisection of $\mathbb{CP}^2$
  4. Bridge trisections
  5. Transverse bridge position
  6. Algebraic transversality and adjunction
  7. Acknowledgements
  8. Diagrams for surfaces in $\mathbb{CP}^2$
  9. Trisection Diagram for $\mathbb{CP}^2$
  10. Handlebody coordinates
  11. Orientations
  12. Surfaces in $\mathbb{CP}^2$
  13. Torus diagrams of surfaces in $\mathbb{CP}^2$
  14. Degree formulas
  15. Bridge stabilization
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\mathcal{K}$ be a smoothly embedded, oriented, connected surface in $\mathbb{CP}^2$ of degree $d > 0$. Then

Figures (18)

  • Figure 1: The moment polytope of $\mathbb{CP}^2$, with the trisection decomposition described.
  • Figure 2: ( Left) A torus diagram for a bridge trisection of a cubic curve in $\mathbb{CP}^2$. ( Right) A banded link diagram corresponding to the bridge splitting of the cubic.
  • Figure 3: An example piece torus diagram. Crossing 1 contributes $-1$ to the writhe $w(K_3)$. Crossing 2 contributes $+1$ to the writhe $w(K_1)$. Crossing 3 contributes $+1$ to $w(K_1)$ and $-1$ to $w(K_3)$.
  • Figure 4: (Left) A positive and negative bridge point in the shadow diagram. (Right) A positively twisted, untwisted, and negatively twisted band in the shadow diagram.
  • Figure 5: A torus diagram depiction of a mini bridge stabilization.
  • ...and 13 more figures

Theorems & Definitions (71)

  • Theorem 1.1: Thom Conjecture KM-Thom
  • Theorem 1.2: Ribbon-Bennequin inequality
  • Theorem 1.3: Local Thom Conjecture KM-Milnor
  • Theorem 1.4
  • Definition 1.5: Gay-Kirby-Trisections
  • Proposition 1.6
  • Definition 1.7
  • Theorem 1.8: MZ-GBT
  • Theorem 1.9
  • Theorem 1.11
  • ...and 61 more