Relations amongst twists along Montesinos twins in the 4-sphere

TL;DR

This paper investigates the subgroup of the smooth mapping class group of the 4-sphere generated by twists along Montesinos twins, showing it is either trivial or cyclic of order two, thus clarifying its structure.

Contribution

It establishes that the subgroup generated by twists along Montesinos twins in the 4-sphere is either trivial or of order two, providing new insights into 4-sphere diffeomorphism classes.

Findings

The subgroup from loops of 5-dimensional handles coincides with twists along Montesinos twins.

This subgroup is proven to be either trivial or cyclic of order two.

The result clarifies the structure of the smooth mapping class group of the 4-sphere.

Abstract

Isotopy classes of diffeomorphisms of the 4-sphere can be described either from a Cerf theoretic perspective in terms of loops of 5-dimensional handle attaching data, starting and ending with handles in cancelling position, or via certain twists along submanifolds analogous to Dehn twists in dimension two. The subgroup of the smooth mapping class group of the 4-sphere coming from loops of 5-dimensional handles of index 1 and 2 coincides with the subgroup generated by twists along Montesinos twins (pairs of 2-spheres intersecting transversely twice) in which one of the two 2-spheres in the twin is unknotted. In this paper we show that this subgroup is in fact trivial or cyclic of order two.