Second obstruction to pseudoisotopy in dimension 3

TL;DR

This paper constructs a nontrivial diffeomorphism of a 4-manifold using the second obstruction to pseudoisotopy, demonstrating it is not isotopic to the identity, thus advancing understanding of 3- and 4-dimensional topology.

Contribution

It introduces a novel application of the second obstruction to pseudoisotopy to produce explicit nontrivial diffeomorphisms in dimension 4.

Findings

Constructed a nontrivial diffeomorphism of M×I using lens-shaped models.

Demonstrated the diffeomorphism of M×S^1 is not isotopic to the identity.

Extended the understanding of obstructions in pseudoisotopy theory.

Abstract

We use lens-shaped models and the second obstruction to pseudoisotopy to construct a nontrivial diffeomorphism of $M\times I$ where $M$ is the connected sum of $S^1\times S^2$ with a another nonsimply connected 3-manifold $M'$. Then we take two copies of this diffeomorphism and paste together their tops and bottoms to obtain a diffeomorphism of $M\times S^1$. Properties of the second obstruction and the first Postnikov invariant imply that this diffeomorphism of the closed 4-manifold $M\times S^1$ is not isotopic to the identity. Similar results were obtain by Singh [10].