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

TL;DR

This paper investigates the relationship between families of spheres and circles in certain 4-manifolds, showing how they induce nontrivial diffeomorphisms of a 3-manifold product, and establishes the infinite complexity of these diffeomorphism classes.

Contribution

It demonstrates the injection of fundamental groups from circle families to sphere families and proves the nontriviality and distinctness of associated barbell diffeomorphisms.

Findings

Countably many nontrivial sphere families from circle families

Diffeomorphisms induced are nontrivial and pairwise distinct

Infinite generation of the isotopy classes group of diffeomorphisms

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.