The 4-dimensional light bulb theorem

Gabai, David

doi:10.1090/jams/920

The 4-dimensional light bulb theorem
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by David Gabai

J. Amer. Math. Soc. 33 (2020), 609-652

DOI: https://doi.org/10.1090/jams/920

Published electronically: June 15, 2020

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Abstract:

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. 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 $\mathbb {Z}_2$-torsion, one surface can be put into a normal form relative to the other.

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