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The 4-dimensional light bulb theorem


Author: David Gabai
Journal: J. Amer. Math. Soc. 33 (2020), 609-652
MSC (2000): Primary 57N13, 57N35
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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Additional Information

David Gabai
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: gabai@math.princeton.edu

DOI: https://doi.org/10.1090/jams/920
Received by editor(s): July 14, 2017
Received by editor(s) in revised form: July 15, 2018, December 21, 2018, and January 31, 2019
Published electronically: June 15, 2020
Additional Notes: This research was partially supported by NSF grants DMS-1006553, 1607374
Article copyright: © Copyright 2020 David Gabai