Diagonal complexes

Abstract

It is known that the partially ordered set of all tuples of pairwise non- intersecting diagonals in an -gon is isomorphic to the face lattice of a convex polytope called the associahedron. We replace the -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 (generalizing the associahedron) with the barycentric subdivision . When the surface is closed, the complex (as well as ) is homotopy equivalent to the space of metric ribbon graphs or, equivalently, to the decorated moduli space . 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.