On Watanabe's theta graph diffeomorphism in the 4-sphere

David T Gay

Vol. 340 (2026), No. 2, 229–244

DOI: 10.2140/pjm.2026.340.229

Abstract

Watanabe’s theta graph diffeomorphism, constructed using Watanabe’s clasper surgery construction which turns trivalent graphs in $4$-manifolds into parametrized families of diffeomorphisms of $4$-manifolds, is a diffeomorphism of $S^4$ representing a potentially nontrivial smooth mapping class of $S^4$. The “$(1,2)$-subgroup” of the smooth mapping class group of $S^4$ is the subgroup represented by diffeomorphisms which are pseudoisotopic to the identity via a Cerf family with only index $1$ and $2$ critical points. This author and Hartman showed that this subgroup is either trivial or has order $2$ and explicitly identified a diffeomorphism that would represent the nontrivial element if this subgroup is nontrivial. Here we show that the theta graph diffeomorphism is isotopic to this one possibly nontrivial element of the $(1,2)$-subgroup. To prove this relation we develop a diagrammatic calculus for working in the smooth mapping class group of $S^4$.

Keywords

diffeomorphism, 4-sphere, barbell, pseudoisotopy

Mathematical Subject Classification

Primary: 57K40

Secondary: 57R50, 57R52, 57R65

Milestones

Received: 9 December 2024

Revised: 19 September 2025

Accepted: 24 September 2025

Published: 26 November 2025