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Separating incompressible surfaces and stabilizations of Heegaard splittings. (English) Zbl 1062.57028
The authors develop an approach for the description of 3-manifolds containing closed separating incompressible surfaces of arbitrary large genus and find a “simplest” manifold of such sort. This result provides two applications.
1. For a closed orientable 3-manifold \(M\) and any positive integer \(m\) the surgery along any link \(L\) in \(M\) with at most \(2m+1\) components provides an irreducible 3-manifold containing \(m\) disjoint non-parallel separaring surfaces of arbitrarily high genus.
2. There exist 3-manifold \(M\) containing separating incompressible surfaces \(S_n\) of arbitrarily large genera \(g(S_n)\) such that the amalgamation of minimal Heegaard splittings of two resulting 3-manifolds cutting along \(S_n\) can be stabilized \(g(S_n)-3\) times to a minimal Heegaard splitting of \(M\).

57N10 Topology of general \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
57N16 Geometric structures on manifolds of high or arbitrary dimension
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