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Relative Thom Conjectures, symplectic and beyond

Matthew Hedden, Katherine Raoux

TL;DR

This work generalizes genus-minimization from closed holomorphic and symplectic surfaces to relative settings by introducing the Relative Almost Complex Thom Conjecture and the Relative Symplectic Thom Conjecture for cobordisms to 3-manifolds. It develops a toolkit—self-linking numbers, the relative adjunction formula, and a genus relation—to relate boundary data to interior genus and uses knot Floer-theoretic tau invariants to produce obstructions to links bounding symplectic surfaces in fillings and symplectizations. The main results connect Floer-theoretic boundary data to relative genus gaps, yielding corollaries such as the relative Thom conjecture, tau-based obstructions, and half-symplectization consequences, with implications toward a symplecto-geometric understanding of tightness. The work thus broadens Thom-type phenomena beyond purely holomorphic or symplectic settings and suggests new interactions between contact topology, Floer theory, and symplectic geometry.

Abstract

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate corollary affirms the relative symplectic Thom conjecture and, moreover, yields obstructions coming from knot Floer homology to a link bounding a symplectic surface in a symplectic filling. Our results are applicable to knots in manifolds equipped with plane fields that admit no symplectic fillings; for instance, we show that symplectic surfaces in a thickening of any contact 3-manifold with non-zero Ozsvath-Szabo invariant minimize slice genus for their boundary. We conjecture that this phenomenon occurs precisely when the contact structure is tight, which would imply that tightness can be viewed as a symplecto-geometric notion.

Relative Thom Conjectures, symplectic and beyond

TL;DR

This work generalizes genus-minimization from closed holomorphic and symplectic surfaces to relative settings by introducing the Relative Almost Complex Thom Conjecture and the Relative Symplectic Thom Conjecture for cobordisms to 3-manifolds. It develops a toolkit—self-linking numbers, the relative adjunction formula, and a genus relation—to relate boundary data to interior genus and uses knot Floer-theoretic tau invariants to produce obstructions to links bounding symplectic surfaces in fillings and symplectizations. The main results connect Floer-theoretic boundary data to relative genus gaps, yielding corollaries such as the relative Thom conjecture, tau-based obstructions, and half-symplectization consequences, with implications toward a symplecto-geometric understanding of tightness. The work thus broadens Thom-type phenomena beyond purely holomorphic or symplectic settings and suggests new interactions between contact topology, Floer theory, and symplectic geometry.

Abstract

We establish a criterion that ensures a bounded almost complex curve in a bounded almost complex 4-manifold minimizes genus amongst all smooth surfaces that share its homology class and the transverse link on its boundary. An immediate corollary affirms the relative symplectic Thom conjecture and, moreover, yields obstructions coming from knot Floer homology to a link bounding a symplectic surface in a symplectic filling. Our results are applicable to knots in manifolds equipped with plane fields that admit no symplectic fillings; for instance, we show that symplectic surfaces in a thickening of any contact 3-manifold with non-zero Ozsvath-Szabo invariant minimize slice genus for their boundary. We conjecture that this phenomenon occurs precisely when the contact structure is tight, which would imply that tightness can be viewed as a symplecto-geometric notion.

Paper Structure

This paper contains 5 sections, 14 theorems, 38 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Relative adjunction and genus formulae
  3. Self-linking numbers
  4. Relative adjunction formulas
  5. The relative almost complex Thom conjecture

Key Result

Theorem 1

If $\Sigma$ is a smoothly embedded symplectic surface in a closed symplectic $4$-manifold $(W,\omega)$, then for any smoothly embedded $\Sigma'\subset W$ with $[\Sigma]=[\Sigma']\in H_2(W)$.

Figures (1)

  • Figure 1: The top sequence shows $X$ winding once counterclockwise with respect to $s_\nu$. The bottom sequence is the same, but viewed as $s_\nu$ winding with respect to $X$ --- from this perspective $s_\nu$ winds once clockwise.

Theorems & Definitions (30)

  • Theorem 1: Symplectic Thom Conjecture SympThom
  • Theorem 2: Relative Almost Complex Thom Conjecture
  • Theorem 3
  • Corollary 4: Relative Symplectic Thom Conjecture
  • Theorem 5
  • Remark 1.1
  • Corollary 6
  • Conjecture 7: Tightness is symplectically determined
  • Proposition 2.1: The Relative Adjunction Formula
  • proof
  • ...and 20 more