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A note on bicompressible surface and unstabilized amalgamated Heegaard splitting. (English) Zbl 1411.57026
This paper gives sufficient conditions for Heegaard splittings, which are obtained by operations like amalgamation, to be unstabilized or to be uncritical.
Let $$F$$ be a compact orientable separating surface properly embedded in a compact orientable 3-manifold $$M$$. If $$F$$ is bicompressible, then the (Hempel) distance $$d(F)$$ can be defined by using its curve complex as in [J. Hempel, Topology 40, No. 3, 631–657 (2001; Zbl 0985.57014)].
Suppose that $$F$$ has non-empty boundary components. We further assume that $$M$$ is irreducible and $$\partial$$-irreducible, and that closing $$F$$ by attaching sub-surfaces of $$\partial M$$ gives a Heegaard surface, say $$S$$, for $$M$$. Then this paper shows that such a Heegaard surface $$S$$ is unstabilized if $$d(F)\ge 6$$.
Suppose next that $$F$$ is closed and non-separating and that there is a surface $$F'$$ such that $$F\cup F'$$ separates $$M$$ into two pieces, say $$M_1$$ and $$M_2$$. Let $$S_i$$ $$(i=1,2)$$ be a Heegaard surface in $$M_i$$ such that $$S_i$$ cuts off a handlebody, say $$V_i$$, from $$M_i$$. We further assume that the closure of $$M_2\setminus V_2$$ has only a single essential disk. Then the Heegaard surface obtained by amalgamating $$S_1\cup S_2$$ is unstabilized and is uncritical if $$d(S_1)\ge 3$$.
##### MSC:
 57M27 Invariants of knots and $$3$$-manifolds (MSC2010) 57M50 General geometric structures on low-dimensional manifolds 57N10 Topology of general $$3$$-manifolds (MSC2010)
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