A necessary and sufficient condition for a Heegaard splitting to be keen weakly reducible (Q2087776)

Heegaard splittings are an important topic in low dimensional topology and play an important role in the study of \(3\)-manifolds. For example, \textit{J. Hempel} [Topology 40, No. 3, 631--657 (2001; Zbl 0985.57014)] defined the Heegaard distance by using the curve complex and proved that there exist arbitrarily high distance Heegaard splittings for \(3\)-manifolds. \textit{R. C. S. Bernardo} and \textit{J. P. H. Esguerra} [Ann. Phys. 391, 293--311 (2018; Zbl 1384.81063)] defined keen Heegaard splittings and proved the existence of strongly keen Heegaard splittings; \textit{L. Liang} et al. [Topology Appl. 237, 1--6 (2018; Zbl 1382.57011)] gave a sufficient condition for weakly keen Heegaard splittings to be keen. In the paper under review, the authors give a necessary and sufficient condition for a Heegaard splitting of a compact orientable \(3\)-manifold to be keen weakly reducible as follows: \textbf{Theorem 1.1.} Suppose \(V\cup_{S}W\) is an irreducible Heegaard splitting for an orientable closed \(3\)-manifold \(M\) with \(g(S) > 2\), then \(V\cup_{S}W\) is keen weakly reducible if and only if every pair in \((\mathcal{D}_{V}, \mathcal{D}_{W})\) realizing the Hempel distance \(1\) is a \(W\)-pair and has no \(d\)-strong intersection property. \textbf{Theorem 1.2.} Suppose \(C_{1}\cup_{S} C_{2}\) is an irreducible Heegaard splitting for a compact orientable \(3\)-manifold \(M\) that has non-sphere boundary components with \(g(S) > 2\). Then \(C_{1}\cup_{S} C_{2}\) is keen weakly reducible if and only if any pair \((x, y)\in (\mathcal{D}_{C_{1}}, \mathcal{D}_{C_{2}})\) that can realize the Hempel distance \(1\) and has the \(d\)-strong intersection property is either a \(W\)-pair or an \(S\)-pair. And if there exists an \(S\)-pair which can realize the Hempel distance \(1\), then it is unique.