On the Lego-Teichmüller game
Let \(\Sigma\) be an oriented surface, possibly with boundary. For the study of its mapping class group \(\Gamma(\Sigma)\) or for the construction of a modular functor it is convenient to subdivide \(\Sigma\) into spheres with holes. Such a cut system can always be found and generally more than one. The product of the mapping class groups of the pieces then maps to \(\Gamma(\Sigma)\), and indeed one can get this way a full set of generators and relations of \(\Gamma(\Sigma)\). This is the approach taken by \textit{A. E. Hatcher} and \textit{W. Thurston} [Topology 19, 221-237 (1980; Zbl 0447.57005)]. This approach was also suggested by \textit{W. Grothendieck} [Lond. Math. Soc. Lect. Note Ser. 242, 5-48, 243-283 (1997; Zbl 0901.14001)] who called it the `Lego-Teichmüller game'. A modular functor \(V\) -- as studied in conformal field theory -- assigns to each surface \(\Sigma\) a vector space \(V(\Sigma)\) such that homeomorphisms give rise to linear isomorphisms. Modular functors satisfy certain gluing laws so that \(V\) is completely determined by its values on spheres with holes. As there are many cut systems for a given surface, the question arises whether there are some simple moves such that any cut system can be transformed into any other by a sequence of such moves, and given a set of simple moves, how unique such a sequence is. The goal of this paper is to present a connected and simply-connected 2-dimensional CW-complex in which the vertices are cut systems (or more precisely the authors' refined notion of markings), edges represent simple moves between them, and faces represent relations between different sequences of simple moves. This has also been considered by the physicists \textit{G. Moore} and \textit{N. Seiberg} [Commun. Math. Phys. 123, No. 2, 177-254 (1989; Zbl 0694.53074)]. The authors, however, point out some draw-backs of this work and ``give a reformulation and rigorous proof of the result of Moore and Seiberg''.
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