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The 4-Dimensional Light Bulb Theorem

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\BZ_2$-torsion in the fundamental group. This gives a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for a simple closed curve in $S^4$ and $π_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.