Symplectic trisections and the adjunction inequality

TL;DR

This paper proves a version of the adjunction inequality for symplectic 4-manifolds using contact geometry and trisections, avoiding gauge theory and enabling new proofs of key results in 4-manifold topology.

Contribution

It introduces a gauge-theory-free approach to the adjunction inequality via contact geometry and trisections, linking it to the slice-Bennequin inequality and Khovanov homology.

Findings

Establishes a new adjunction inequality for symplectic 4-manifolds.

Provides gauge-theory-free proofs of the Thom conjecture and related results.

Enables detection of exotic smooth structures and restrictions on decompositions.

Abstract

In this paper, we establish a version of the adjunction inequality for closed symplectic 4-manifolds. As in a previous paper on the Thom conjecture, we use contact geometry and trisections of 4-manifolds to reduce this inequality to the slice-Bennequin inequality for knots in the 4-ball. As this latter result can be proved using Khovanov homology, we completely avoid gauge theoretic techniques. This inequality can be used to give gauge-theory-free proofs of several landmark results in 4-manifold topology, such as detecting exotic smooth structures, the symplectic Thom conjecture, and exluding connected sum decompositions of certain symplectic 4-manifolds.