Spaces of Geodesic Triangulations Are Cells

TL;DR

This paper explores the topological structure of spaces of geodesic triangulations on negatively curved surfaces, aiming to prove they are homeomorphic to Euclidean spaces, building on known contractibility results.

Contribution

It introduces an approach to establish that these spaces are homeomorphic to Euclidean spaces, extending previous contractibility findings.

Findings

Spaces of geodesic triangulations are contractible.

Proposed approach to prove homeomorphism to Euclidean spaces.

Potential new topological characterization of triangulation spaces.

Abstract

It has been shown that spaces of geodesic triangulations of surfaces with negative curvature are contractible. Here we propose an approach aiming to prove that the spaces of geodesic triangulations of a surface with negative curvature are homeomorphic to Euclidean spaces $\mathbb{R}^n$.