A topologically extendible mapping class that is not smoothly extendible

TL;DR

This paper constructs examples of topologically but not smoothly extendible mappings of a torus in a specific 4-manifold, highlighting differences between topological and smooth categories in 4-dimensional topology.

Contribution

It provides explicit examples of embeddings and diffeomorphisms that distinguish topological from smooth extendibility in 4-manifolds.

Findings

Existence of a topologically extendible Dehn twist not smoothly extendible.

Examples of topologically isotopic but smoothly non-isotopic embeddings of an annulus.

Illustration of the subtle differences between topological and smooth structures in 4-manifold topology.

Abstract

We give an example of a smooth characteristic embedding of a torus in $\s^2 \times \s^2 \# \s^1 \times \s^3$ such that there exists no diffeomorphism of the ambient $4$-manifold that induces the Dehn twist along a meridian of the torus, but there exists a homeomorphism of the ambient $4$-manifold, isotopic to identity, that induces the Dehn twist. As an application of our methods, we provide examples of two proper smooth embeddings of an annulus in $\s^2 \times \s^2 \# \s^1 \times \s^3 \setminus int(\D^4)$ which are topologically isotopic, but not smoothly isotopic (relative to boundary).