Spaces of geodesic triangulations of the sphere

Abstract

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