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.Comment: Included into section 2 of this version is a proof that the operation
of `sliding a Whitney disk over itself' preserves the isotopy class of the
resulting Whitney move in the current setting. Some expository clarifications
have also been added. Main results and proofs are unchanged from the previous
version. 39 pages, 25 figure
