# Diagonal complexes

## TL;DR

This paper generalizes the associahedron to arbitrary oriented surfaces with marked points, establishing homotopy equivalences with moduli spaces and providing combinatorial models for tautological bundles and psi-classes.

## Contribution

It introduces diagonal complexes for surfaces, relates them to moduli spaces, and models tautological bundles combinatorially, extending associahedron concepts.

## Key findings

- Homotopy equivalence of diagonal complexes with moduli spaces for closed surfaces.
- Boundary contractions model tautological circle bundles.
- Combinatorial formulas for psi-classes and bundle sums.

## Abstract

Given an $n$-gon, the poset of all collections of pairwise non-crossing diagonals is isomorphic to the face poset of some convex polytope called \textit{associahedron}. We replace in this setting the $n$-gon (viewed as a disc with $n$ marked points on the boundary) with an arbitrary oriented surface with a number of labeled marked points ("vertices"). With appropriate definitions we arrive at cell complexes $\mathcal{D}$ (generalization of) and its barycentric subdivision $\mathcal{BD}$. The complex $\mathcal{D}$ generalizes the associahedron. If the surface is closed, the complex $\mathcal{D}$ (as well as $\mathcal{BD}$) 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 a boundary edge does not change the homotopy type of the support 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's tautological circle bundle. Thus, contraction of a boundary component gives 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. The latter result is an application of the local combinatorial formula. (3) In the same way, contraction of several boundary components corresponds to Whitney sum of the tautological bundles.