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Destabilizing amalgamated Heegaard splittings. (English) Zbl 1216.57011
Gordon, Cameron (ed.) et al., Proceedings of the Technion workshop on Heegaard splittings, Haifa, Israel, summer 2005. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 12, 319-334 (2007).
Summary: We construct a sequence of pairs of 3-manifolds $$(M_1^n,M_2^n)$$ each with incompressible torus boundary and with the following two properties: (1) For $$M^n$$ the result of a carefully chosen glueing of $$M_1^n$$ and $$M_2^n$$ along their boundary tori, the genera $$(g_1^n, g_2^n)$$ of $$(M_1^n, M_2^n)$$ and the genus $$g^n$$ of $$M^n$$ satisfy the inequality $$g^n/(g_1^n+ g_2^n)<1/2$$. (2) The result of amalgamating certain unstabilized Heegaard splittings of $$M_1^n$$ and $$M_2^n$$ to form a Heegaard splitting of $$M$$ produces a stabilized Heegaard splitting that can be destabilized successively $$n$$ times.
For the entire collection see [Zbl 1133.57002].

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010) 57M27 Invariants of knots and $$3$$-manifolds (MSC2010)
##### Keywords:
Heegaard genus; 3-manifolds; destabilizations
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