Translator’s Introduction 8

Abstract

The final paper in this series is in effect a vindication of Dehn’s 1922 Breslau lecture, although it did not appear until 1938. In it Dehn studies mapping classes by their action on simple curve systems, for orientable surfaces of arbitrary genus p, finding that they are represented by linear transformations of the space of (6p-6)-dimensional integer vectors, and giving a finite set of generators, which represent twist maps (now known as Dehn twists). The proof is by induction on the complexity of the surface and the mapping, with a very complicated base step which requires detailed analysis of spheres with up to 5 holes. Perhaps because of its very demanding proof, the result went unnoticed until it was rediscovered independently by Lickorish [1962]. His proof is similar in concept to Dehn’s but much shorter, using twists to reduce complexity until a very simple base step is reached. Some further simplifications are made in Birman [1977]. An impressive aspect of Lickorish’s rediscovery is that he made it in conjunction with a rediscovery of Dehn surgery, when Dehn himself had not connected these two ideas!