For any topological manifold this bijection need to be a homeomorphism, if not the result will not be a topological manifold. Equally for your differentiable manifold it has to be a diffeomorphism. For other manifolds other buildings needs to be preserved. As the valves are full bore by having an http://shaneuelrz.free-blogz.com/17845952/rumored-buzz-on-manifold-valve