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
\(\bullet\) \(d(P) \leq 2 \text{ genus}(Q)\).


57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
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