On the Lego-Teichmuller game

For a smooth oriented surface Sigma, denote by M(Sigma) the set of all ways to represent Sigma as a result of gluing together standard spheres with ho les ("the Lego game"). In this paper we give a full set of simple moves and relations which turn M(Sigma) into a connected and simply-connected 2-comp lex. Results of this kind were first obtained by Moore and Seiberg, but the ir paper contains serious gaps. Our proof is based on a different approach and is much more rigorous.