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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 $$M$$.

MSC:
 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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