On 3-manifolds with Torus or Klein Bottle Category Two

Abstract A subset W of a closed manifold M is K-contractible, where K is a torus or Klein bottle if the inclusion W → M factors homotopically through a map to K. The image of π1(W) (for any base point) is a subgroup of π1(M) that is isomorphic to a subgroup of a quotient group of π1(K). Subsets of M with this latter property are called 𝒢K-contractible. We obtain a list of the closed 3-manifolds that can be covered by two open 𝒢K-contractible subsets. This is applied to obtain a list of the possible closed prime 3-manifolds that can be covered by two open K-contractible subsets.