Diffeomorphisms of the 4-sphere, Cerf theory and Montesinos twins

TL;DR

This paper explores the structure of the smooth mapping class group of the 4-sphere by constructing generators as twists along Montesinos twins, linking loop groups of 2-spheres in $S^2 imes S^2$ to this group.

Contribution

It introduces a surjective homomorphism from a loop group of 2-spheres to the 4-sphere's mapping class group and identifies generators as twists along Montesinos twins.

Findings

Identified a surjective homomorphism onto the mapping class group.

Described generators as twists along Montesinos twins.

Proved one subgroup is in the kernel of the homomorphism.

Abstract

One way to better understand the smooth mapping class group of the 4-sphere would be to give a list of generators in the form of explicit diffeomorphisms supported in neighborhoods of submanifolds, in analogy with Dehn twists on surfaces. As a step in this direction, we describe a surjective homomorphism from a group associated to loops of 2-spheres in $S^2 \times S^2$'s onto this smooth mapping class group, discuss two natural and in some sense complementary subgroups of the domain of this homomomorphism, show that one is in the kernel, and give generators as above for the image of the other. These generators are described as twists along Montesinos twins, i.e. pairs of embedded 2-spheres in the 4-sphere intersecting transversely at two points.