Spaces of Geodesic Triangulations of the Sphere

Abstract

We study questions concerning the homotopy-type of the space $\operatorname {GT} (K)$ of geodesic triangulations of the standard $n$-sphere which are (orientation-preserving) isomorphic to $K$. We find conditions which reduce this question to analogous questions concerning spaces of simplexwise linear embeddings of triangulated $n$-cells into $n$-space. These conditions are then applied to the $2$-sphere. We show that, for each triangulation $K$ of the $2$-sphere, certain large subspaces of $\operatorname {GT} (K)$ are deformable (in $\operatorname {GT} (K)$) into a subsapce homeomorphic to $\operatorname {SO} (3)$. It is conjectured that (for $n = 2$) $\operatorname {GT} (K)$ has the homotopy of $\operatorname {SO} (3)$. In a later paper the authors hope to use these same conditions to study the homotopy type of spaces of geodesic triangulations of the $n$-sphere, $n > 2$.