For a topological manifold this bijection need to be a homeomorphism, usually the result will not be a topological manifold. Equally for the differentiable manifold it should be a diffeomorphism. For other manifolds other buildings really should be preserved. Non-singular algebraic types above the real or complicated quantities are manifolds. http://codynajqx.tinyblogging.com/manifold-valve-for-Dummies-28384139