The 4-dimensional light bulb theorem

By David Gabai. Journal of The American Mathematical Society, 2017

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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 $\pi_0(\Diff_0(S^2\times D^2)/\Diff_0(B^4))=1$. In manifolds with $\BZ_2$-torsion, one surface can be put into a normal form relative to the other.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/30469b99626a8bdefafefb16be7c8343b0577019