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.

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.