Diagonal complexes

It is known that the partially ordered set of all tuples of pairwise non-intersecting diagonals in an n-gon is isomorphic to the face lattice of a convex polytope called the associahedron. We replace the n-gon (viewed as a disc with n marked points on the boundary) by an arbitrary oriented surface with a set of labelled marked points (vertices). After appropriate definitions we arrive at a cell complex D (generalizing the associahedron) with the barycentric subdivision BD. When the surface is closed, the complex D (as well as BD) is homotopy equivalent to the space RGg,nmet of metric ribbon graphs or, equivalently, to the decorated moduli space fMg,n. For bordered surfaces we prove the following. 1) Contraction of an edge does not change the homotopy type of the complex. 2) Contraction of a boundary component to a new marked point yields a forgetful map between two diagonal complexes which is homotopy equivalent to the Kontsevich tautological circle bundle. Thus we obtain a natural simplicial model for the tautological bundle. As an application, we compute the psiclass, that is, the first Chern class in combinatorial terms. This result is obtained by using a local combinatorial formula. 3) In the same way, contraction of several boundary components corresponds to the Whitney sum of tautological bundles.