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.
