# Homotopy versus isotopy: spheres with duals in 4-manifolds

## TL;DR

This paper extends Gabai's 4D Light Bulb Theorem to manifolds with arbitrary fundamental groups by using an invariant that characterizes when homotopy implies isotopy for embedded 2-spheres with duals.

## Contribution

It introduces a new invariant-based criterion for homotopy implying isotopy in 4-manifolds with complex fundamental groups, generalizing previous results.

## Key findings

- Invariant fully characterizes homotopy vs. isotopy for 2-spheres with duals
- Application to unknotting numbers and pseudo-isotopy classes
- Alternative proof of Gabai's theorem using Whitney disks

## Abstract

David Gabai recently proved a smooth 4-dimensional "Light Bulb Theorem" in the absence of 2-torsion in the fundamental group. We extend his result to 4-manifolds with arbitrary fundamental group by showing that an invariant of Mike Freedman and Frank Quinn gives the complete obstruction to "homotopy implies isotopy" for embedded 2-spheres which have a common geometric dual. The invariant takes values in an Z/2Z-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudo-isotopy classes of self-diffeomorphisms. Our methods also give an alternative approach to Gabai's theorem using various maneuvers with Whitney disks and a fundamental isotopy between surgeries along dual circles in an orientable surface.