The Kirby torus trick for surfaces

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.