The Heegaard distances cover all nonnegative integers (Q2346696)

It is well-known that every closed orientable 3-manifold admits a Heegaard splitting, that is, a splitting into two handlebodies, which plays an important role in the study of 3-manifolds. To study properties of Heegaard splittings, in [\textit{J. Hempel}, Topology 40, No. 3, 631--657 (2001; Zbl 0985.57014)], Hempel introduced a complexity of Heegaard splittings, now called the \textit{(Hempel) distance} of a Heegaard splitting. In that paper, Hempel exhibited Heegaard splittings of distance \(n\) for arbitrarily large \(n\). In the paper under review, it is proved that, for any integers \(n \geq 1\) and \(g \geq 2\), there is a closed 3-manifold which admits a Heegaard splitting of genus \(g\) and of distance \(n\) unless \((g, n) = (2, 1)\), and the manifolds can be chosen to be hyperbolic unless \((g, n) = (3, 1)\) (Theorem 1.1). Furthermore, also is shown that, for any integers \(g \geq 2\) and \(n \geq 4\), there are infinitely many non-homeomorphic closed 3-manifolds admitting Heegaard splittings of genus \(g\) and distance \(n\) (Theorem 1.3). Note that similar results have been independently obtained by A. Ido, Y. Jang, and T. Kobayashi [\textit{A. Ido} et al., Algebr. Geom. Topol. 14, No. 3, 1395--1411 (2014; Zbl 1297.57029)].