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 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 is homotopy equivalent to the space of metric ribbon graphs $RG_{g,n}^{met}$, 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.