# The 4-Dimensional Light Bulb Theorem

## TL;DR

This paper generalizes the light bulb trick to 4-dimensional manifolds, establishing conditions under which embedded 2-spheres are isotopic and exploring related uniqueness and normal form results.

## Contribution

It extends classical 3D techniques to 4D, providing new isotopy criteria and normal form results for embedded spheres in 4-manifolds.

## Key findings

- Embedded 2-spheres with the same transverse sphere are isotopic under certain conditions
- Uniqueness of spanning discs for simple closed curves in S^4
- The fundamental group torsion condition affects normal form results

## Abstract

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.