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

TL;DR

This paper investigates Watanabe's theta graph diffeomorphism in the 4-sphere, showing it is isotopic to a potentially nontrivial element of a specific subgroup of the smooth mapping class group, using a new diagrammatic calculus.

Contribution

The paper develops a diagrammatic calculus for the smooth mapping class group of S^4 and relates Watanabe's theta graph diffeomorphism to a known subgroup element.

Findings

Watanabe's theta graph diffeomorphism is isotopic to a potentially nontrivial element of the (1,2)-subgroup.

A new diagrammatic calculus for the smooth mapping class group of S^4 is introduced.

The theta graph diffeomorphism may represent a nontrivial smooth mapping class of S^4.

Abstract

Watanabe's theta graph diffeomorphism, constructed using Watanabe's clasper surgery construction which turns trivalent graphs in 4-manifolds into parameterized 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 $S^4$.