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 F2-vector space generated by elements of order 2 in the fundamental group and has applications to unknotting numbers and pseudoisotopy classes of selfdiffeomorphisms. 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.
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