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Alternate Heegaard genus bounds distance. (English) Zbl 1128.57022
Summary: Suppose $$M$$ is a compact orientable irreducible $$3$$-manifold with Heegaard splitting surfaces $$P$$ and $$Q$$. Then either $$Q$$ is isotopic to a possibly stabilized copy of $$P$$ or the distance $$d(P) \leq 2 \text{ genus}(Q)$$.
More generally, if $$P$$ and $$Q$$ are bicompressible but weakly incompressible connected closed separating surfaces in $$M$$ then either
$$\bullet$$ $$P$$ and $$Q$$ can be well-separated or
$$\bullet$$ $$P$$ and $$Q$$ are isotopic or
$$\bullet$$ $$d(P) \leq 2 \text{ genus}(Q)$$.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds
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##### References:
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