Relations amongst twists along Montesinos twins in the 4-sphere

Gay, David T; Hartman, Daniel

doi:10.2140/agt.2025.25.287

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Relations amongst twists along Montesinos twins in the 4-sphere

David T Gay and Daniel Hartman

Algebraic & Geometric Topology 25 (2025) 287–299

DOI: 10.2140/agt.2025.25.287

Abstract

Isotopy classes of diffeomorphisms of the $4$ -sphere can be described either from a Cerf-theoretic perspective in terms of loops of $5$ -dimensional handle attaching data, starting and ending with handles in canceling position, or via certain twists along submanifolds analogous to Dehn twists in dimension $2$ . The subgroup of the smooth mapping class group of the $4$ -sphere coming from loops of $5$ -dimensional handles of index $1$ and $2$ coincides with the subgroup generated by twists along Montesinos twins (pairs of $2$ -spheres intersecting transversely twice) in which one of the two $2$ -spheres in the twin is unknotted. We show that this subgroup is in fact trivial or cyclic of order $2$ .

Keywords

pseudoisotopy, Montesinos twins, diffeomorphism, isotopy, 4-manifold, mapping class group

Mathematical Subject Classification

Primary: 57K40

Secondary: 57K45

References

Publication

Received: 4 April 2023

Revised: 10 September 2023

Accepted: 23 September 2023

Published: 5 March 2025

Authors

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

Daniel Hartman
Department of Mathematics University of Georgia Athens, GA United States

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Volume 25, issue 1 (2025)