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The Heegaard genus of amalgamated 3-manifolds. (English) Zbl 1081.57018
Genera of closed $$3$$-manifolds obtained from two $$3$$-manifolds by glueing along their boundaries are studied. Let $$g(X)$$ denote the genus of $$X$$. Let $$M_1, M_2$$ be simple $$3$$-manifolds. Suppose that there is a homeomorphism $$h_i : \partial M_i \rightarrow S$$ for $$i=1, 2$$, where $$S$$ is connected and of genus at least two. Let $$\psi : S \rightarrow S$$ be a pseudo-Anosov homeomorphism. The main theorem states that $$g(M_1 \cup_{\Psi} M_2) = g(M_1) + g(M_2) - g(S)$$, if $$\Psi = h_2^{-1} \psi^n h_1$$ and $$| n|$$ is sufficiently large. Furthermore, any minimal genus Heegaard splitting for $$M_1 \cup_{\Psi} M_2$$ is obtained from splittings of $$M_1$$ and $$M_2$$ by amalgamation and is weakly reducible. Amalgamation was defined by J. Schultens [Proc. London Math. Soc. (3) 67, 425–448 (1993; Zbl 0789.57012)]. This holds even for $$\Psi=h_2^{-1} \phi h_1$$ with $$\phi$$ of sufficiently large distance. For general glueing homeomorphisms, certain inequalities have been discovered by K. Johannson [Topology and Combinatorics of $$3$$-Manifolds, Springer-Verlag (Berlin) (1995; Zbl 0820.57001)] and J. Schultens [Heegaard genus formula for Haken manifolds, preprint]. Genera of non-minimal Heegaard splittings, and that of $$M/\sim$$ obtained by glueing two components of $$\partial M$$ are also studied. We summarise the proof of the main theorem. $$M_1 \bigcup_{\Psi} M_2$$ has a hyperbolic structure by the geometrisation theorem. There are an arbitrarily large number of parallel copies of $$S$$ in $$M_1 \cup_{\Psi} M_2$$ such that any two adjacent ones have distance at least one from each other, provided that $$| n|$$ is sufficiently large. Untelescoping a minimal genus Heegaard splitting $$F$$, we construct a generalised Heegaard splitting. The positive splitting surfaces $$F_+$$ may be isotoped in a small neighbourhood of a minimal surface. Then we can find a copy of $$S$$ disjoint from $$F_+$$. It is parallel to some component of the generalised splitting surfaces, and hence $$F$$ is an amalgamation of Heegaard splittings of $$M_1$$ and $$M_2$$.

##### MSC:
 57N10 Topology of general $$3$$-manifolds (MSC2010)
##### Keywords:
Haken manifold; minimal surfaces; hyperbolic geometry
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##### References:
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