Pseudo-isotopy versus isotopy for homeomorphisms of 4-manifolds

TL;DR

This paper introduces obstructions that distinguish pseudo-isotopies from isotopies in 4-manifolds, matching known smooth obstructions and demonstrating their realizability through specific homeomorphisms.

Contribution

It defines new obstructions for topological pseudo-isotopies in 4-manifolds, aligning with smooth obstructions, and constructs examples of homeomorphisms that are pseudo-isotopic but not isotopic.

Findings

Obstructions match smooth obstructions of Hatcher-Wagoner

Obstructions are fully realizable

Constructs homeomorphisms pseudo-isotopic but not isotopic

Abstract

We define obstructions which obstruct topological pseudo-isotopies from being isotopic to isotopies in dimension four. These match the smooth obstructions of Hatcher-Wagoner for smooth pseudo-isotopies, and accordingly are valued in certain Whitehead groups. We show that our obstructions are fully realisable, and we use these realisations to build homeomorphisms of $Y\times S^1$ for many 3-manifolds $Y$ that are pseudo-isotopic to the identity but not isotopic to the identity.