For each composite number n ̸ = 2 k, there does not exist a single connected closed (n + 1)-manifold such that any smooth, simply-connected, closed n-manifold can be topologically flat embedded into it. There is a single connected closed 5-manifold W such that any simply-connected, 4-manifold M can be topologically flat embedded into W if M is either closed and indefinite, or compact and with non-empty boundary. 1 Introduction and some prerequisites The celebrated Whitney embedding theorem states that any smooth n-dimensional manifold can be embedded smoothly into the Euclidean space R 2n, or equivalently, any smooth n-dimensional manifold can be embedded into the sphere S 2n, the simplest closed 2n-manifold. If the target space is allowed to be othe
