On keen weakly reducible Heegaard splittings (Q2411460)

Critical surfaces and topologically minimal surfaces were introduced by \textit{D. Bachman} [Trans. Am. Math. Soc. 354, No. 10, 4015--4042 (2002; Zbl 1003.57023); Geom. Topol. 14, No. 1, 585--609 (2010; Zbl 1206.57020)], respectively. They are very useful in the theory of Heegaard splittings of \(3\)-manifolds. For example, \textit{D. Bachman} proved Gordon's Conjecture in [Geom. Topol. 12, No. 4, 2327--2378 (2008; Zbl 1152.57020)] by using critical Heegaard surfaces. By the definition of critical surface, if the Heegaard surface of a Heegaard splitting is critical, then the Heegaard splitting is weakly reducible. By the definition of topological index of a surface \(F\), \(F\) has topological index \(0\) if and only if it is incompressible, and has topological index \(1\) if and only if it is strongly irreducible. \textit{A. Ido, Y. Jang} and \textit{T. Kobayashi} defined keen Heegaard splittings of \(3\)-manifolds in [``On keen Heegaard splittings'', Preprint, \url{arXiv:1605.04513}] and showed that for any integers \(n\geq 2\) and \(g\geq 3\), there exists a strongly keen Heegaard splitting of genus \(g\) whose distance is \(n\). In the paper under review, the author considers keen weakly reducible Heegaard surfaces and whether a keen weakly reducible Heegaard surface is critical or topologically minimal as follows: Theorem 3.1. For each \(g>2\), there are infinitely many genus \(g\), keen, weakly reducible Heegaard splittings of closed \(3\)-manifolds. Corollary 4.1. Suppose that a \(3\)-manifold \(M\) admits an amalgamated Heegaard splitting: \(M=V\cup_{S} W=(V_{1}\cup_{S_{1}} W_{1})\cup_{F}(V_{2}\cup_{S_{2}} W_{2})\), such that \(g(F)=g(S_{1})=g(S_{2})\) and \(d(S_{i})\geq 4\) for \(i=1, 2\), then \(V\cup_{S} W\) is keen weakly reducible and \(S\) is not topologically minimal. Corollary 4.2. Suppose that a \(3\)-manifold \(M\) admits an amalgamated Heegaard splitting: \(M=V\cup_{S} W=(V_{1}\cup_{S_{1}} W_{1})\cup_{F}(V_{2}\cup_{S_{2}} W_{2})\), such that \(g(F)=g(S_{1})=g(S_{2})\) and \(d(S_{i})\geq 2\) for \(i=1, 2\), then \(S\) is either critical or not topologically minimal.