The Kirby torus trick for surfaces

TL;DR

This paper presents a simplified proof of the existence and uniqueness of smooth structures on topological surfaces using the Kirby torus trick, avoiding complex point-set topology.

Contribution

It introduces a new proof method leveraging the Kirby torus trick, simplifying the understanding of smooth structures on surfaces.

Findings

Existence of smooth structures on topological surfaces

Uniqueness of smooth structures up to isotopy

Reduction of topological complexity in proofs

Abstract

This is an expository paper giving a proof of the existence and uniqueness of smooth structures (hence also PL structures) on topological surfaces. Most published proofs rely on the topological Schoenflies theorem, but here we use instead the Kirby torus trick. This has the advantage of reducing the point-set topology in the proof to practically nothing, replacing it by a few basic facts about smooth surfaces. Uniqueness of smooth structures is proved in the strong form that every homeomorphism between smooth surfaces is isotopic to a diffeomorphism.