Band diagrams of immersed surfaces in 4-manifolds

TL;DR

This paper develops a diagrammatic method for representing and classifying immersed surfaces in 4-manifolds, introducing singular banded unlink diagrams and bridge trisections, with applications to computations and homotopies.

Contribution

It introduces singular banded unlink diagrams for immersed surfaces in 4-manifolds and proves their uniqueness and applicability to homotopies and bridge trisections.

Findings

Every self-transverse immersed surface can be represented by a singular banded unlink diagram.

Such diagrams are uniquely determined up to specific singular band moves.

The methods enable explicit computations and homotopies of surfaces.

Abstract

We study immersed surfaces in smooth 4-manifolds via singular banded unlink diagrams. Such a diagram consists of a singular link with bands inside a Kirby diagram of the ambient 4-manifold, representing a level set of the surface with respect to an associated Morse function. We show that every self-transverse immersed surface in a smooth, orientable, closed 4-manifold can be represented by a singular banded unlink diagram, and that such representations are uniquely determined by the ambient isotopy or equivalence class of the surface up to a set of singular band moves which we define explicitly. By introducing additional finger, Whitney, and cusp diagrammatic moves, we can use these singular band moves to describe homotopies or regular homotopies as well. Using these techniques, we introduce bridge trisections of immersed surfaces in arbitrary trisected 4-manifolds and prove that such bridge trisections exist and are unique up to simple perturbation moves. We additionally give some examples of how singular banded unlink diagrams may be used to perform computations or produce explicit homotopies of surfaces.