On Topology of G-Configuration Spaces of Polyhedra

The family P of all convex 3-polytopes P in Euclidean space E 3 may be partitioned into combinatorial types or configuration spaces by isomorphism of face lattices , and the configuration space [ ] P of any such 3-polytope P may be subdivided further into GConfiguration space P by equivalence of actions of symmetry group ) (P G on face lattices. With respect to a natural topology induced by Hausdorff metric on P, each [ ] P is a contractible manifold of a certain dimension related to the number of edges of P . This is a consequence of Steinitz’s fundamental theorem of convex polyhedra.. In this article we prove an extension of this theorem which states that if P is a convex 3-polytopes with symmetry group ( ) P G , then the G -Configuration space P is also an smooth manifold. The dimension , dim P of this manifold is calculated with respect to the orbits of the action of symmetry group ( ) P G on face lattice of P .