×

Local detection of strongly irreducible Heegaard splittings. (English) Zbl 0926.57018

Let \(M\) be a compact \(3\)-manifold containing a Heegaard surface \(S\) that splits \(M\) into two handlebodies \(H_1\) and \(H_2\) (or, if \(M\) has nonempty boundary, into two compression bodies). A. J. Casson and C. McA. Gordon [ibid. 27, No. 3, 275-283 (1987; Zbl 0632.57010)] introduced the notion of a strongly irreducible Heegaard splitting. This means that any pair of essential discs, one in \(H_1\) and one in \(H_2\), must have intersecting boundaries. Such splittings are prevalent. For example, in a non-Haken \(3\)-manifold, any splitting either is strongly irreducible or is reducible, and consequently any minimal genus splitting is strongly irreducible.
In this paper, the author shows that for a strongly irreducible Heegaard splitting, the possible intersections of \(S\) with a \(3\)-ball or a solid torus are significantly constrained. If \(B\) is a \(3\)-ball in \(M\) (transverse to \(S\)) for which the planar surfaces \(\partial B\cap H_i\) are incompressible, then \(S\cap B\) is connected, planar, and unknotted in \(B\) (the latter means that it is parallel to a submanifold of \(\partial B\)). If \(V\) is a solid torus such that \(\partial V\) intersects \(S\) in parallel essential curves, not bounding discs in \(V\), then \(S\) intersects \(V\) in a collection of boundary-parallel annuli and possibly one other component, obtained from one or two annuli by attaching a tube along an arc parallel to a subarc of \(\partial V\). If the latter sort of component is in \(V\), then \(S-V\) is incompressible in \(M-V\).

MSC:

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bonahon, F.; Otal, J. P., Scindements de Heegaard des espaces lenticulaires, Ann. Sci. École Norm. Sup., 16, 451-466 (1983) · Zbl 0545.57002
[2] Casson, A.; Gordon, C. McA., Reducing Heegaard splittings, Topology Appl., 27, 275-283 (1987) · Zbl 0632.57010
[3] Frohman, C., The topological uniqueness of triply periodic minimal surfaces in \(R^3\), J. Differential Geom., 31, 277-283 (1990) · Zbl 0689.53002
[4] Haken, W., Some results on surfaces in 3-manifolds, (Studies in Modern Topology (1968), Prentice-Hall), 34-98 · Zbl 0194.24902
[6] Scharlemann, M.; Thompson, A., Thin position and Heegaard splittings of the 3-sphere, J. Differential Geom., 39, 343-357 (1994) · Zbl 0820.57005
[7] Scharlemann, M.; Thompson, A., Thin position for 3-manifolds, (Contemp. Math., 164 (1994), Amer. Math. Soc: Amer. Math. Soc Providence, RI), 231-238 · Zbl 0818.57013
[8] Waldhausen, Heegaard-Zerlegungen der 3-Sphäre, Topology, 7, 195-203 (1968) · Zbl 0157.54501
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.