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 $\mathcal{D}$ (generalizing the associahedron) with the barycentric subdivision $\mathcal{BD}$. When the surface is closed, the complex $\mathcal{D}$ (as well as $\mathcal{BD}$) is homotopy equivalent to the space $RG_{g,n}^{\mathrm{met}}$ of metric ribbon graphs or, equivalently, to the decorated moduli space $\widetilde{\mathcal{M}}_{g,n}$. 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 psi-class, 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.