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

TL;DR

The paper proves that Watanabe’s theta graph diffeomorphism $\mathrm{wat}(\Theta)$ in $S^4$ is equal to the Cerf-theory–based element $\mathcal{H}(\alpha(1))$ in the mapping class group $\mathrm{Mod}(S^4)$, situating both within the $(1,2)$-subgroup $\mathrm{Mod}^{1,2}(S^4)$. It develops a diagrammatic toolbox to compare these two diffeomorphisms by representing 1-parameter families of embeddings and surgeries as a 6-dimensional handle diagram, and then shows through a concrete sequence of diagrammatic moves—handle cancellations, slides, isotopies, and a splitting of spinning spheres—that wat$\big(\Theta\big)$ and $\mathcal{H}(\alpha(1))$ correspond to the same element in $\mathrm{Mod}(S^4)$. The work provides a new, diagrammatic route to relate Watanabe’s clasper-surgery constructions to the Cerf-theoretic approach to pseudoisotopies, and reinforces the known result that $\mathrm{Mod}^{1,2}(S^4)$ is generated by $\mathcal{H}(\alpha(1))$ with order at most $2$, yielding an alternative proof aligned with Kosanović’s Corollary. The techniques offer a flexible diagrammatic calculus for exploring relations in the smooth mapping class groups of 4-manifolds and may illuminate further structure in $\mathrm{Mod}(S^4)$.

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 $S^4$.

Figures (11)

• Figure 1: The beginning and the end of our forthcoming sequence of diagrams. Section \ref{['S:Diagrams']} explains the meaning of the two diagrams in this figure, Section \ref{['S:BeginningAndEnd']} justifies the left-most and right-most equals signs, and Section \ref{['S:Sequence']} gives a sequence of intermediate diagrams justifying the middle equals sign.
• Figure 2: A spinning sphere as a $t$--parameterized family of arcs, for $t \in [1/4,3/4]$. In the top line we first draw the figure in the $3$--dimensional slice $\{x_4=0\}$, so that the sphere appears as a red circle with three linking black arcs, and we then draw the sphere in the slice $\{x_3=0\}$ so that we see the entire sphere, but the arcs now appear as three black dots.
• Figure 3: A finger move along an arrow and back.
• Figure 4: Watanabe's construction applied to the theta graph.
• Figure 5: $\mathop{\rm wat}\nolimits(\Theta)$ with labelled handles.

Theorems & Definitions (2)

• Theorem 1
• proof : Proof of Theorem \ref{['T:Main']}