S2- and P2-category of manifolds

A closed topological n-manifold M n is of S 2 (resp. P 2 )-category 2 if it can be covered by two open subsets W 1 , W 2 such that the inclusions W i → M n factor homotopically through maps W i → S 2 (resp. P 2 ). We characterize all closed n-manifolds of S 2 -category 2 and the fundamental groups of all closed n-manifolds of P 2 -category 2.