Homotopy versus isotopy: Spheres with duals in 4-manifolds

Abstract

Dave 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 ${\mathbb{F}}_2$-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudoisotopy 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.