A 4-dimensional light bulb theorem for disks

TL;DR

This paper extends the 4-dimensional light bulb theorem to properly embedded disks, providing new conditions for isotopy and a geometric interpretation of the Dax invariant, advancing understanding in 4-manifold topology.

Contribution

It generalizes the 4D light bulb theorem for disks, offering new isotopy criteria and a geometric perspective on the Dax invariant.

Findings

Homotopic disks with a common dual are smoothly isotopic under certain conditions.

Provides a new geometric interpretation of the Dax invariant.

Extends previous theorems from 2-spheres to disks in 4-manifolds.

Abstract

We give a 4-dimensional light bulb theorem for properly embedded disks, generalizing recent work of Gabai and Kosanovic-Teichner in certain contexts, and extending the 4-dimensional light bulb theorem for 2-spheres due to Gabai and Schneiderman-Teichner. In particular, we provide conditions under which homotopic disks properly embedded in a compact 4-manifold X with a common dual in the interior of X are smoothly isotopic rel boundary. We also provide a new geometric interpretation of the Dax invariant, to aid in its computation.