An adjunction criterion in almost-complex 4-manifolds

TL;DR

This paper introduces a new criterion based on polyhedral decompositions for applying the adjunction inequality to surfaces in a broad class of almost-complex 4-manifolds, extending previous gauge-theoretic methods.

Contribution

It develops a novel polyhedral decomposition approach to establish adjunction inequalities in almost-complex 4-manifolds without relying on global geometric invariants.

Findings

Provides a new adjunction criterion for almost-complex 4-manifolds.

Extends the applicability of adjunction inequalities beyond symplectic cases.

Connects polyhedral decompositions with gauge-theoretic invariants.

Abstract

The adjunction inequality is a key tool for bounding the genus of smoothly embedded surfaces in 4-manifolds. Using gauge-theoretic invariants, many versions of this inequality have been established for both closed surfaces and surfaces with boundary. However, these invariants generally require some global geometry, such as a symplectic structure or nonzero Seiberg-Witten invariants. In this paper, we extend previous work on trisections and the Thom conjecture to obtain adjunction information in a much larger class of smooth 4-manifolds. We intrdouce polyhedral decompositions of almost-complex 4-manifolds and give a criterion in terms of this decomposition for surfaces to satisfy the adjunction inequality.