Uniqueness of surface diagrams of smooth 4-manifolds

TL;DR

This paper proves that surface diagrams of smooth 4-manifolds are uniquely determined up to four moves when derived from maps within the same homotopy class, enhancing understanding of 4-manifold representations.

Contribution

It establishes a uniqueness theorem for surface diagrams of smooth 4-manifolds, showing they are unique up to four specific moves within a fixed homotopy class.

Findings

Surface diagrams are uniquely determined up to four moves.

The moves include stabilization, handleslide, multislide, and shift.

This result clarifies the classification of 4-manifold surface diagrams.

Abstract

In the author's earlier work there appeared a new way to specify any smooth closed 4-manifold by a surface diagram, which consists of an orientable surface decorated with simple closed curves. These curves are cyclically indexed, and each curve has a unique transverse intersection with the next. Each surface diagram comes from a certain type of map from the 4-manifold to the two-sphere. The aim of this paper is to give a uniqueness theorem stating that surface diagrams coming from maps within a fixed homotopy class are unique up to four moves: stabilization, handleslide, multislide, and shift.