Diffeomorphisms of 4-manifolds from graspers

TL;DR

This work builds a unified framework connecting degree-one graspers to fundamental 4-manifold diffeomorphism constructions via a parameterised surgery map $\mathsf{ps}_{\nu S}$. It shows Watanabe's theta classes, Budney–Gabai barbell diffeomorphisms, and Gay twists arise as images of graspers under $\mathsf{ps}$, with explicit identifications such as $\mathsf{wat}(\Theta) = \mathsf{ps}(\mathfrak{r}^{\circlearrowright}_{\circlearrowright}(h))$ for suitable $h\in\pi_1$; in particular for $M=\mathbb{S}^4$ these constructions collapse to a single 2-torsion class $W(1)$ (i.e., $W(1)^2=\mathrm{Id}$) that coincides with $\mathsf{ps}(\mathfrak{r}(t))$ and with $\mathsf{bg}(\mathcal{b}_{\mathsf{y}\mathsf{x}})$. The paper also develops the Dax invariant as an explicit inverse to the grasping map on its image, clarifying kernels and lifts to framed embeddings, and articulates how framed-embedding data interacts with pseudo-isotopy via a parameterised handle-cancellation construction. Overall, it provides explicit, computable bridges between classical 4-manifold diffeomorphism constructions and the graspers framework, yielding concrete formulas for a broad class of diffeomorphisms and demonstrating the central role of the 2-torsion phenomenon in the $\mathbb{S}^4$ case.

Abstract

We relate degree one grasper families of embedded circles to various constructions of diffeomorphisms found in the literature -- theta clasper classes of Watanabe, barbell diffeomorphisms of Budney and Gabai, and twin twists of Gay and Hartman. We use a ``parameterised surgery map'' that for a smooth 4-manifold $M$ takes loops of framed embeddings of $S^1$ in the manifold obtained by surgery on some 2-sphere in $M$, to the mapping class group of $M$.

Figures (18)

• Figure 1: The embedded torus $T_{\mathfrak{r}(h)}\colon\mathbb{S}^1\times\mathbb{S}^1\hookrightarrow M\#\mathbb{S}^1\times\mathbb{S}^3$ is the connect-sum of a thin torus containing the blue $c=\mathbb{S}^1\times\{pt\}$, with a meridian sphere for $c$, along a guiding arc going around $h\in\pi_1(M\#\mathbb{S}^1\times\mathbb{S}^3)$. The class $\mathsf{ps}({\mathfrak{r}(h)})\in\pi_0\mathop{\mathrm{Diff}}\nolimits(M)$ is the Gay twist $G(\Sigma_{\mathfrak{r}(h)})$ on the circle bundle $\Sigma_{\mathfrak{r}(h)}=\partial(\nu T_{\mathfrak{r}(h)})$, and then surger $c$ out. Roughly speaking, $G(\Sigma_{\mathfrak{r}(h)})$ does Dehn twists on the curves that guide the movement of $c$ around $T_{\mathfrak{r}(h)}$. The 2-torsion class $\mathsf{ps}({\mathfrak{r}(h)})\in\pi_0\mathop{\mathrm{Diff}}\nolimits(\mathbb{S}^4)$ is the case $M=\mathbb{S}^4$ and $h=\mathsf{c}$.
• Figure 2: (i) To get $\mathfrak{r}(h)_t\colon\mathbb{S}^1\hookrightarrow X$ connect-sum the tip of the blue finger into each of the red arcs, that foliate a meridian sphere to $c$. (ii) The simple grasper for $\mathfrak{r}(h)$. (iii) The semisimple grasper for $\mathfrak{r}^\circlearrowright_\circlearrowright(h)$.
• Figure 3: (i) The barbell $\delta_m\coloneqq\mathcal{b}_{g\mathsf{y} g^{m-3}\mathsf{x} g^2}$ in $\mathbb{D}^3\times\mathbb{S}^1$ is the thickening of the two cuffs and the bar connecting them. By putting $g=1$ we view this as the barbell $\mathcal{b}_{\mathsf{y}\mathsf{x}}$ in $\mathbb{D}^4\subset\mathbb{S}^4$. (ii) Another picture of $\delta_m\subset\mathbb{D}^3\times\mathbb{S}^1$, with $g\in\mathbb{Z}\cong\pi_1(\mathbb{D}^3\times\mathbb{S}^1)$ as the $y$-axis.
• Figure 4: (i) The family $\mathfrak{r}(-h)$. (ii) The family $\mathfrak{r}(h_1+h_2)$.
• Figure 5: (i) The family $\mathfrak{r}(h)$. (ii) The circle in this family completely contained in the present, $\{0\}\times\mathbb{R}^3$. (ii) The only immersed circle in the family $\mathfrak{r}(h)_{s,\tau}$, and the local picture around the double point.

Theorems & Definitions (68)

• Theorem 1: Theorem \ref{['thm:Theta']}, Corollary \ref{['cor:Wat-implant']}
• Corollary 1.1: Corollary \ref{['cor:implant-D3xS1']}
• Corollary 1.2
• Corollary 1.3
• Corollary 1.4: Corollary \ref{['cor:implant-formula']}
• Definition 1.5
• Definition 1.6
• Theorem 2
• proof
• Remark 1.7