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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.