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.