Spaces of Geodesic Triangulations of Surfaces

TL;DR

This paper proves the contractibility of the space of geodesic triangulations for convex polygons and shows that for any n, there exists a triangulation space with non-trivial n-th homotopy group, revealing complex topological structures.

Contribution

It provides a short proof of contractibility for geodesic triangulation spaces and demonstrates the existence of non-trivial higher homotopy groups in these spaces.

Findings

Contractibility of geodesic triangulation space for convex polygons

Existence of spaces with non-trivial higher homotopy groups

Topological complexity of triangulation spaces

Abstract

We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $n>0$, we show that there exists a space of geodesic triangulations of a polygon with a triangulation, whose $n$-th homotopy group is not trivial.