Spaces of Geodesic Triangulations of Surfaces

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.

16 pages, 10 figures. Comments are very welcomed!