Pseudo-isotopies and diffeomorphisms of 4-manifolds

TL;DR

This paper explores the distinction between pseudo-isotopies and isotopies of diffeomorphisms in 4-manifolds, providing new examples and understanding of pseudo-isotopy obstructions.

Contribution

It constructs examples of pseudo-isotopic but not isotopic diffeomorphisms of 4-manifolds and analyzes the realizability of pseudo-isotopy obstructions.

Findings

Examples of pseudo-isotopic but not isotopic diffeomorphisms in 4-manifolds.

Realization of all first and second pseudo-isotopy obstructions after connected sums with S^2×S^2.

Deeper understanding of the second pseudo-isotopy obstruction in 4-manifolds.

Abstract

A diffeomorphism $f$ of a compact manifold $X$ is pseudo-isotopic to the identity if there is a diffeomorphism $F$ of $X\times I$ which restricts to $f$ on $X\times 1$, and which restricts to the identity on $X\times 0$ and $\partial X\times I$. We construct examples of diffeomorphisms of 4-manifolds which are pseudo-isotopic but not isotopic to the identity. To do so, we further understanding of which elements of the "second pseudo-isotopy obstruction", defined by Hatcher and Wagoner, can be realised by pseudo-isotopies of 4-manifolds. We also prove that all elements of the first and second pseudo-isotopy obstructions can be realised after connected sums with copies of $S^2\times S^2$.