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Surfaces, submanifolds, and aligned Fox reimbedding in non-Haken 3-manifolds. (English) Zbl 1071.57015
R. H. Fox showed that any compact connected $$3$$-dimensional submanifold $$M$$ of the $$3$$-sphere $$S^3$$ is homeomorphic to the complement of a disjoint union of handlebodies in $$S^3$$ [Ann. Math. (2) 49, 462–470 (1948; Zbl 0032.12502)]. This is generalized in the paper under review.
Let $$N$$ be a closed orientable irreducible non-Haken $$3$$-manifold, and $$M$$ be a connected compact $$3$$-submanifold of $$N$$ with non-empty boundary. Then $$M$$ is homeomorphic to a submanifold of $$N$$ whose complement is homeomorphic to a disjoint union of handlebodies or to $$N \sharp$$ (disjoint union of handlebodies), where $$\sharp$$ denotes connected sum. Moreover, when the boundary $$\partial M$$ is connected, keeping the above conditions, $$M$$ can be reimbedded in $$N$$ so that there is a complete collection of reducing spheres $${\mathbb S}$$ for $$\partial M$$ with $$M-{\mathbb S}$$ being a union of solid tori and $$\partial$$-irreducible manifolds. A sphere in $$N$$ is called a reducing sphere for $$\partial M$$ if it intersects $$\partial M$$ in an essential circle. A collection of reducing spheres $${\mathbb S}$$ is called complete if the number of non-planar components of $$\partial M - {\mathbb S}$$ is no less than that for any other collection containing $${\mathbb S}$$.
In the course of the proof, the theorem below is shown. Let $$H$$ be a handlebody of genus $$g$$ imbedded in a closed orientable irreducible non-Haken $$3$$-manifold $$N$$. If $$\partial H$$ is compressible in the complement of $$H$$, then either (1) the Heegaard genus of $$N$$ is less than or equal to $$g$$, (2) $$\partial H$$ has a reducing sphere or (3) $$H$$ has an unknotted core. The last condition (3) means that there are a proper disk $$D$$ in $$H$$ and a circle $$c$$ in $$\partial H$$ such that $$c$$ intersects $$\partial D$$ in a single point and $$c$$ bounds a disk in $$N$$ transverse to $$\partial H$$.

##### MSC:
 57M50 General geometric structures on low-dimensional manifolds 57M25 Knots and links in the $$3$$-sphere (MSC2010)
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##### References:
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