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 \(M-V\).


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M99 General low-dimensional topology
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