Smoothing topological pseudo-isotopies of 4-manifolds

TL;DR

This paper investigates when topologically pseudo-isotopic diffeomorphisms of 4-manifolds are also smoothly pseudo-isotopic, establishing affirmative results for certain fundamental groups and providing the first counterexamples.

Contribution

It proves conditions under which topological pseudo-isotopies imply smooth pseudo-isotopies and constructs the first examples where this implication fails.

Findings

For fundamental groups in certain classes, topological and smooth pseudo-isotopies coincide.

Constructs the first known examples where they differ.

Addresses an open question about smooth isotopy versus topological isotopy in 4-manifolds.

Abstract

Given a closed, smooth 4-manifold $X$ and self-diffeomorphism $f$ that is topologically pseudo-isotopic to the identity, we study the question of whether $f$ is moreover smoothly pseudo-isotopic to the identity. If the fundamental group of $X$ lies in a certain class, which includes trivial, free, and finite groups of odd order, we show the answer is always affirmative. On the other hand, we produce the first examples of manifolds $X$ and diffeomorphisms $f$ where the answer is negative. Our investigation is motivated by the question, which remains open, of whether there exists a self-diffeomorphism of a closed 4-manifold that is topologically isotopic to the identity, but not stably smoothly isotopic to the identity.