Spaces of geodesic triangulations of the sphere

Abstract

We study questions concerning the homotopy-type of the space 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 GT(K) are deformable (in GT(/f)) into a subsapce homeomorphic to SO(3). It is conjectured that (for n = 2) GT(/f) has the homotopy of 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.