On reduction complexity of Heegaard splittings (Q1916449)

Let \(M\) be a closed connected orientable 3-manifold. For any Heegaard splitting \(V \cup_F W\) of \(M\) a reduction complexity \(\delta\) of \(V \cup_F W\) is introduced. It is shown that if the genus of \(V \cup_F W\) is \(n\), then (1) \(\delta \geq \max\{2,n\}\) iff \(V \cup_F W\) is reducible, (2) \(\delta \geq 2\) iff \(V \cup_F W\) is weakly reducible, and (3) \(\delta > n\) iff \(M\) has exactly \(\delta - n\) connected sum factors \(S^1\times S^2\). Moreover, the following refinement of a result of \textit{A. J. Casson} and \textit{C. McA. Gordon} [Topology Appl. 27, 275-283 (1987; Zbl 0632.57010)] is obtained: if \(V \cup_F W\) is a genus \(n\) Heegaard splitting of \(M\) and \(n > \delta \geq 2\), then \(M\) contains an incompressible surface of genus \(n - \delta\).