Heegaard splittings of Haken manifolds have bounded distance.
A closed 3-manifold \(M\) is called Haken if it is irreducible (i.e. every embedded 2-sphere bounds a ball) and admits an essential surface (i.e. an embedded incompressible surface). A Heegaard splitting for \(M\) is given by an embedded surface \(\Sigma\subset M\) so that \(M\) cut open along \(\Sigma\) consists of 2 handlebodies, say \(V_1\) and \(V_2\). Given two simple closed curves (say \(\alpha\) and \(\beta\)) embedded in some surface, the distance between \(\alpha\) and \(\beta\) is defined to be the smallest \(n\) such that there exists a chain of simple closed curves on the given surface \(\alpha= \gamma_0,\gamma_1,\dots, \gamma_n= \beta\) with the condition that \(\gamma_{i-1}\cap \gamma_1=\emptyset\) \((i= 1,\dots, n)\). The distance associated with a Heegaard splitting is the minimal distance between curves \(\alpha\) and \(\beta\) where \(\alpha\) bounds a disk in \(V_1\) and \(\beta\) bounds a disk in \(V_2\). The author of the current paper proves that if a Haken manifold contains an essential surface of genus \(g\) then the distance of any Heegaard splitting is at most \(2g\). The manifold \(M\) may have non-empty boundary but the surface considered must be closed and not boundary parallel. This generalizes a well-known theorem of Haken that says that if a manifold is reducible (i.e, contains an essential surface of genus \(0\)) then any Heegaard splitting is reducible (which is equivalent to having distance zero). The technique of this theorem is useful in other situations (cf. [\textit{D. Bachman}, Topology Appl. 116, No. 2, 153--184 (2001); erratum ibid. 143, No. 1--3 (2004; Zbl 1054.57023)] and [\textit{M. Scharleman} and \textit{M. Tomova}, Alternate Heegard genus bounds distance, UCSB 2004-38] for recent results). We briefly describe it here: a Heegaard splitting gives rise to a height function (or sweepout) in a well understood way. This induces a height function on the essential surface \(S\). For very low heights, \(S\) provides a compressing disk for the level surface below the level. For a very high level, a compression above the level exists. To bound the distance between the boundaries of these disks, we consider the saddles between S and the various levels. Note that at each saddle one curve breaks to two (or vice versa). Roughly speaking, only when all three curves are essential, the collection of essential curves on the surface can change. Since a neighborhood of a saddle in the surface \(S\) is a pair of pants, a Euler characteristic argument shows that there can be no more than \(2g\) such saddles. This gives rise to a chain of curves of length no more than \(2g+ 1\).
scientific article; zbMATH DE number 1818984