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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 \(M_2\).

57N10 Topology of general \(3\)-manifolds (MSC2010)
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