An obstruction to smooth isotopy in dimension 4

TL;DR

This paper introduces a gauge-theoretic invariant for diffeomorphisms of 4-manifolds, revealing a new obstruction to smooth isotopy and providing the first example of a diffeomorphism homotopic but not smoothly isotopic to the identity.

Contribution

It defines a novel gauge-theoretic invariant that detects non-smooth isotopy in 4-manifolds, demonstrating its effectiveness with a new example.

Findings

Invariant vanishes for smoothly isotopic diffeomorphisms

Constructs the first example of a homotopic but not smoothly isotopic diffeomorphism in a simply-connected 4-manifold

Shows gauge theory techniques can distinguish smooth structures in 4D

Abstract

Techniques of gauge theory are used to define and compute an invariant of certain diffeomorphisms of 4-manifolds. The invariant vanishes for any diffeomorphism which is smoothly isotopic to the identity. As an application, we give the first example of a diffeomorphism of a simply-connected 4-manifold which is homotopic to the identity map, but not smoothly isotopic to the identity.