Universal links for \(S^2\widetilde\times S^1\) (Q1392483)

There are main results as follows: There exists a five component link \(U \subset S^2\times S^1\) such that every closed, connected, orientable 3-manifold \(M\) with \(H^1 (M)\neq 0\) is a branched covering over \(S^2 \times S^1\) with branching set exactly the link \(U\). There exists a five component link \(U\subset S^2 \otimes S^1\) such that every closed, connected, non-orientable 3-manifold \(M\) with the Bockstein of the first Stiefel-Whitney class, \(\beta w_1(M)=0\), is a branched covering over \(S^2 \otimes S^1\) branched along the link \(U\).