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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
##### Keywords:
Heegaard splitting; strongly irreducible; solid torus
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##### References:
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