Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

Gay, David T

doi:10.2140/agt.2025.25.2817

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ISSN (electronic): 1472-2739

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Diffeomorphisms of the $4$-sphere, Cerf theory and Montesinos twins

David T Gay

Algebraic & Geometric Topology 25 (2025) 2817–2849

DOI: 10.2140/agt.2025.25.2817

Abstract

One way to better understand the smooth mapping class group of the $4$ -sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces. As a step in this direction, we describe a surjective homomorphism from a group associated to loops of $2$ -spheres in $S^{2} \times S^{2}$ ’s onto this smooth mapping class group, discuss two natural and in some sense complementary subgroups of the domain of this homomorphism, show that one is in the kernel, and give generators as above for the image of the other. These generators are described as twists along Montesinos twins, ie pairs of embedded $2$ -spheres in $S^{4}$ intersecting transversely at two points.

Keywords

diffeomorphism, $4$-sphere, Cerf theory, pseudoisotopy

Mathematical Subject Classification

Primary: 57K40

Secondary: 57R40, 57R65

References

Publication

Received: 4 April 2023

Revised: 31 July 2024

Accepted: 8 September 2024

Published: 17 September 2025

Authors

David T Gay
Department of Mathematics University of Georgia Athens, GA United States

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Volume 25, issue 5 (2025)