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Comparing open book and Heegaard decompositions of 3-manifolds. (English) Zbl 1045.57008

An open book decomposition of a compact 3-manifold \(M\) is a collection of embedded circles \({\mathcal C}\subset M\), so that \(M\setminus{\mathcal C}\) fibres over \(S^1\); moreover, we require that the boundary of each fiber is exactly \(\mathcal C\) (these fibers are called leaves). A Heegaard decomposition of a compact 3-manifold is a decomposition of the 3-manifold into two handlebodies (more generally, compression bodies). The author of the paper under review considers the two structures, and first comments that given an open book decomposition one can obtain a Heegaard splitting: the boundary of a normal neighborhood of a leaf is a Heegaard surface. The genus of that surface is the Euler characteristic of the leaf plus 1. The author then continues and asks whether the Heegaard surface thus obtained is of minimal genus.
In Theorem 2.1 the author considers the following situation: suppose a genus 2 3-manifold has an open book decomposition yielding a minimal genus splitting. This implies very strong constraints on the manifold. In conclusion, the Heegaard splitting induced by an open book decomposition of “most” genus 2 3-manifolds is not minimal genus.
Next (Theorem 3.1), the author considers cusped hyperbolic 3-manifolds (i.e., hyperbolic 3-manifolds that are the interior of a compact 3-manifold whose boundary consists of tori). By Dehn filling a cusped manifold we mean attaching solid tori to the boundary tori. It is a well studied operation that yields infinitely many manifolds. Given any cusped hyperbolic manifold, it is shown that either (1) the Heegaard splittings obtained from open book decompositions of generic filled manifolds have unbounded genera, or (2) the Heegaard splittings obtained from open book decompositions of generic filled manifolds are obtained by adding trivial handles to Heegaard splittings for the cusped manifolds. This is in line with work of Moriah-Rubinstein [Y. Moriah and H. Rubinstein, Commun. Anal. Geom. 5, 375–412 (1997; Zbl 0890.57025)] and the reviewer [Y. Rieck, Topology 39, 619–641 (2000; Zbl 0944.57013)].
In Theorem 3.2 the author gives a partial converse to Theorem 2.1, constructing infinitely many manifolds that possess an open book decomposition, so that the Heegaard splittings obtained from these open book decompositions are all of minimal genus.
The paper is concluded with a list of open problems.

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
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