The 4-Dimensional Light Bulb Theorem

David Gabai

Published 2017-05-28Version 1

For embedded 2-spheres in a 4-manifold sharing the same embedded transverse sphere homotopy implies isotopy, provided the ambient 4-manifold has no $\mathbb Z_2$-torsion in the fundamental group. Among other things, this leads to a generalization of the classical light bulb trick to 4-dimensions, the uniqueness of spanning discs for simple closed curves in $S^4$ and $\pi_0(\textrm{Diff}_0(S^2\times D^2)/\textrm{Diff}_0(B^4))=1$.