On DS-diagrams for 3-manifolds of Heegaard genus 2 (Q854377)

The block number \(Bl(M)\) of a closed orientable 3-manifold \(M\) is a new topological invariant introduced by \textit{M. Endoh} and \textit{I. Ishii} [Jap. J. Math., New Ser. 31, 131--156 (2005; Zbl 1081.57017)]. Its definition depends on the concept of DS-diagram with \(E\)-cycle due to \textit{H. Ikeda} [Kobe J. Math 3, 103--112 (1986; Zbl 0651.57009)]. In the first quoted paper the authors proved that the block number \(Bl(M)\) dominates the Heegaard genus \(HG(M)\) for any \(M\neq\mathbb{S}^2\times \mathbb{S}^1\). Other results are: \(Bl(\mathbb{S}^2\times\mathbb{S}^1)= 0\), \(Bl(\mathbb{S}^3)= 1\), \(Bl(L(p, q))= HG(L(p, q)) = 1\), and \(Bl(M)= HG(M)= 2\) for a Seifert fibered space \(M\) having the 2-sphere \(\mathbb{S}^2\) as its base manifold and three exceptional fibers. An open problem arising from that invariant is to verify if \(HG(M)= Bl(M)\) for any \(M\neq\mathbb{S}^3\), \(\mathbb{S}^2\times \mathbb{S}^1\). In the paper under review the author proves that \(HG(M)= Bl(M)= 2\) for any orientable closed 3-manifold \(M\) of Heegaard genus 2, and that \(Bl(M)\leq 4\) for any \(M\) with Heegaard genus 3.