Minimal genus in 4-manifolds with a free circle action

TL;DR

This paper improves the understanding of the minimal complexity of embedded surfaces in certain 4-manifolds with free circle actions, providing a near-complete classification for a broad class of cases.

Contribution

It extends the adjunction inequality to almost all S^1-bundles over non-graph 3-manifolds, enabling complete determination of minimal surface complexity.

Findings

Enhanced adjunction inequality for most S^1-bundles over N

Complete classification of minimal surface complexity in these bundles

Results apply to a large class of 3-manifolds

Abstract

Let N be a closed irreducible 3-manifold and assume N is not a graph manifold. We improve for all but finitely many S^1-bundles M over N the adjunction inequality for the minimal complexity of embedded surfaces. This allows us to completely determine the minimal complexity of embedded surfaces in all but finitely many S^1-bundles over a large class of 3-manifolds.