S2- and P2-category of manifolds

A closed topological n-manifold Mn is of S2 (resp. P2)-category 2 if it can be covered by two open subsets W1, W2 such that the inclusions Wi→Mn factor homotopically through maps Wi→S2 (resp. P2). We characterize all closed n-manifolds of S2-category 2 and the fundamental groups of all closed n-manifolds of P2-category 2.