# Diagonal complexes for surfaces of finite type and surfaces with involution

## TL;DR

This paper investigates diagonal complexes associated with punctured and symmetric surfaces, establishing their homotopy equivalences and describing their topological structures in relation to surgeries and involutions.

## Contribution

It introduces and analyzes diagonal complexes for surfaces with and without involution, proving their homotopy equivalences and describing their topological properties.

## Key findings

- The symmetric diagonal complex is homotopy equivalent to a punctured surface complex.
- Bundles of surfaces with punctures are related to universal curves with holes.
- Diagonal complexes relate to surgeries of the original surfaces.

## Abstract

Two related constructions are studied: (1) The diagonal complex $\mathcal{D}$ and its barycentric subdivision $\mathcal{BD}$ related to a \textit{punctured} oriented surface $F$ equipped with a number of labeled marked points. (2) The symmetric diagonal complex $\mathcal{D}^{inv}$ and its barycentric subdivision $\mathcal{BD}^{inv}$ related to a symmetric (=with an involution) oriented surface $F$ equipped with a number of (symmetrically placed) labeled marked points. Eliminating a puncture gives rise to a bundle whose fibers are homeomorphic to a surgery of the surface $F$. The bundle can be viewed as the "universal curve with holes". The symmetric complex is shown to be homotopy equivalent to the complex of a punctured surface obtained by a surgery of the initial symmetric surface.