Grasper families of spheres in $S^2 \times D^2$ and barbell diffeomorphisms of $S^1\times S^2 \times I$

TL;DR

The paper analyzes Diff_∂(S^1×S^2×I) by connecting it to spaces of framed embeddings via Cerf–Palais fibrations, proving an injective trade from π_1 Emb(νS^1,S^1×D^3;νc) into π_1 Emb(νS^2,S^2×D^2;νS) with cokernel π_1 Emb_∂(νD^2,D^4). It constructs explicit barbell diffeomorphisms and grasper families that generate infinite rank Z^∞ summands in π_0 Diff_∂ and π_1 Emb(νS^2,S^2×D^2), and provides two independent proofs of their nontriviality. A second, obstruction-theoretic approach uses Hatcher–Wagoner Θ to show each φ^{ℬ_k} is pseudo-isotopic to the identity but not isotopic, and that the φ^{ℬ_k} are pairwise distinct, recovering and refining Singh’s results. Together, the work establishes infinite generation of the abelian group of pseudo-isotopy classes in π_0 Diff_∂(S^1×S^2×I) and links embedding-space phenomena to diffeomorphism theory via explicit geometric models.

Abstract

We show that the fundamental group of framed circles in $S^1 \times D^3$ injects into the fundamental group of framed spheres in $S^2\times D^2$, so that the cokernel is the fundamental group of framed neat disks in $D^4$. In particular, grasper families of circles give rise to countably many nontrivial families of spheres. Ambient extensions of either of these two types of families induce the same barbell diffeomorphisms of $S^1\times S^2\times I$. We give two proofs that these diffeomorphisms are nontrivial and pairwise distinct. This implies infinite generation of the abelian group of isotopy classes of diffeomorphisms of $S^1\times S^2\times I$ that are pseudo-isotopic to the identity, recovering a result of Singh.

Figures (12)

• Figure 1: In each of three pictures we have a Hopf link $S\sqcup c\colon S^2\sqcup S^1\hookrightarrow S^4$ in black; its fixed thickening $\nu(S\sqcup c)$ is not drawn. (i) The support of the diffeomorphism $\phi^{\mathcal{B}_k}$ of $S^1\times S^2\times I=S^4\setminus\nu (S\sqcup c)$ is the barbell$\mathcal{B}_k\subset S^1\times S^2\times I$. This is a thickening of the union of the blue spheres $S_1$ and $S_2$ and the bar connecting them (which links $k\geq1$ times with $S$). (ii) The grasper $c\mathcal{G}_k$ is the union of $\mathcal{B}_k$ and the ball $Q_2$ bounded by $S_2$. The family $\lambda^{c\mathcal{G}_k}_t\colon \nu S^1\hookrightarrow S^1\times D^3$ drags the part $\nu c\cap Q_2$ along the bar until it touches $S_1$, then twirls it around $S_1$, and then goes back. (iii) The grasper $S\mathcal{G}_k$ is the union of $\mathcal{B}_k$ and the ball $Q_1$ bounded by $S_1$. The family $\lambda^{S\mathcal{G}_k}_t\colon\nu S^2\hookrightarrow S^2\times D^2$ drags the part $\nu S\cap Q_1$ along the bar until it touches $S_2$, then twirls its around $S_2$, and then goes back.
• Figure 4: The top row depicts the cross-sections $\Delta_s$ of the midball $\Delta\subset\mathbb B$, and the bottom row of its image $b(\Delta)\subset\mathbb B$ under the barbell map.
• Figure 5: The self-referential grasper on $c\colon S^1\hookrightarrow S^1\times D^3$ for $k\geq1$.
• Figure 6: An isotopy of the sphere $L$ in $S^4\setminus\nu (S\sqcup c)$.
• Figure 7: The manifold $S^1\times S^2\times I \# S^2\times D^2$ and the barbells $\widetilde{\mathcal{B}}_k^{\pm}$ in blue.

Theorems & Definitions (41)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 1
• Theorem 2: KT-4dLBT, KT-mcg
• Lemma 3
• proof
• proof : Proof of Theorem \ref{['T:main-diagram']}
• Corollary 4
• Definition 5: Model barbell, model grasper