Component-conservative invertibility of links and Samsara 4-manifolds on 3-manifolds (Q326878)

Summary: Every Dehn surgery manifold of a component-conservatively invertible link is embedded into a closed oriented 4-manifold with the \(Z_{/2}\)-homology of \(S^1\times S^3\), where \(Z_{/2}=Z[\frac{1}{2}] \) is a subring of \(Q\). This 3-manifold and 4-manifold give a typical example of a closed Samsara 4-manifold on an invertible 3-manifold. After observing that not every closed oriented 3-manifold is a Dehn surgery manifold of a component-conservatively invertible link, we shall construct, for every closed oriented 3-manifold \(M\), an analogous compact oriented 4-manifold but only with as boundary a connected sum of 3-tori, which we call a bounded Samsara 4-manifold on \(M\). This construction leads to an open question asking whether every \(M\) is embedded into a compact oriented 4-manifold with the trivial second \(Q\)-homology. We shall investigate a relationship between the \(Z_2\)-rank of the 2-torsion part of the second homology group of a bounded Samsara 4-manifold on \(M\) and the \(Z_2\)-rank of the first \(Z_ 2\)-homology group of \(M\). From a closed Samsara 4-manifold on \(M\), we also obtain a closed oriented 4-manifold with the \(Z_{/2}\)-homology of \(S^4\) where a punctured 3-manifold \(M^0\) of \(M\) is embedded, called a reduced closed Samsara 4-manifold on \(M^0\). A reduced bounded Samsara 4-manifold on any punctured 3-manifold is also discussed in parallel.