A Dual Point of View on the Ribbon Graph Decomposition of Moduli Space

The ribbon graph decomposition of moduli space is a non-compact orbi-cell complex homeomorphic to Mg,n×R>0. This was introduced by Harer-Mumford-Thurston [Har86] and Penner [Pen87], and used to great effect by Kontsevich [Kon92, Kon94] in his proof of the Witten conjecture and his construction of classes in moduli spaces associated to A∞ algebras. In this note, I discuss in some detail the dual version of the ribbon graph decomposition of the moduli spaces of Riemann surfaces with boundary and marked points, which I introduced in [Cos04a], and used in [Cos04b] to construct open-closed topological conformal field theories. This dual version of the ribbon graph decomposition is a compact orbi-cell complex with a natural weak homotopy equivalence to the moduli space. In the case when all of the marked points are on the boundary of the surface, the combinatorics of the cell complex is captured by ribbon graphs, as usual. In the general case, we find a variant of the ribbon graph complex. The idea of the construction is as follows. We use certain partial compactifications N g,h,r,s of the moduli spaces Ng,h,r,s of Riemann surfaces of genus g with h > 0 boundary components, r boundary marked points, and s internal marked points. The partial compactifications we use are closely related to the Deligne-Mumford spaces; we allow Riemann surfaces with a certain kind of singularity, namely nodes on the boundary. The moduli space N g,h,r,s is an orbifold with corners. The boundary is the locus of singular surfaces. Therefore the inclusion Ng,h,r,s →֒ N g,h,r,s is a weak homotopy equivalence of orbispaces. Inside N g,h,r,s is a natural orbi-cell complex Dg,h,r,s, which is the locus where all the irreducible components of the surface are discs (with at most one internal marked point). We show that the map Dg,h,r,s →֒ N g,h,r,s is a weak homotopy equivalence. This shows that Dg,h,r,s ≃ Ng,h,r,s, giving the desired cellular model for Ng,h,r,s. When s = 0, the combinatorics of the cell complex Dg,h,r,0 is governed by standard ribbon graphs. When s > 0, we find a variant type of ribbon graph, which has two types of vertex. In the standard approach, the non-compact orbi-cell decomposition of moduli space gives a chain model for the Borel-Moore homology, or equivalently the cohomology, of moduli space. The chains are given by ribbon graphs, and the differential is given by summing over ways of contracting an edge, to amalgamate two distinct vertices. The approach used here gives a compact orbi-cell complex, which gives a chain model for homology of moduli space. In the case when s = 0, this is precisely the dual of the standard ribbon graph complex. That is, the chains are given by ribbon graphs, and the differential is given by summing over ways of splitting a vertex into two. The main disadvantage of the approach described here, compared to the more traditional approach, is that we do not find a cell complex homeomorphic to moduli space, but instead a space homotopy equivalent.