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Stabilization, amalgamation and curves of intersection of Heegaard splittings. (English) Zbl 1176.57021
This paper treats the stabilization problem for Heegaard splittings, that is, the problem of determining the number of stabilizations needed to make two given Heegaard splittings of a 3-manifold isotopic. Let \(F\) be a closed orientable essential surface in a compact orientable 3-manifold \(M\), that is, \(F\) is incompressible and none of its components is boundary parallel. Suppose \(F\) is mutually separating, that is, \(M\) cut along \(F\) consists of two (possibly disconnected) 3-manifolds \(X\) and \(Y\) such that every neighborhood of each component of \(F\) intersects both \(X\) and \(Y\) (if \(F\) is not mutually separating, by adding parallel copies of some components of \(F\) we get a mutually separating surface \(F'\)). Given Heegaard splittings of \(X\) and \(Y\), there is a natural way to associate a Heegaard splitting to \(M\), called an amalgamation along \(F\). Let \(V\cup_S W\) be a Heegaard splitting of a 3-manifold \(M\) and let \(F\) be a mutually separating essential surface in \(M\).
In the main result of this paper it is proved that if \(S\) and \(F\) intersect in \(k\) simple closed curves then \(V\cup_S W\) is isotopic to an amalgamation along \(F\) after at most \(k-\chi(F)\) stabilizations. This result can be used to give lower bounds to the number of curves of intersection between a Heegaard surface and an essential surface. The difference between the Heegaard genus of a 3-manifold \(M\) and the genus of an amalgamated splitting can be arbitrarily large, in fact J. Schultens and R. Weidmann [Geom. Topol. Monogr. 12, 319–334 (2007; Zbl 1216.57011)] have shown that for each positive integer \(n\) there exists 3-manifolds \(X_n\) and \(Y_n\) with torus boundary \(T_n\), admitting minimal Heegaard splittings of genus \(n\) and \(n+1\) respectively, so that for certain gluing \(M_n=X_n\cup_{T_n} Y_n\), an amalgamation of these splittings along \(T_n\) gives a genus \(2n\) splitting of \(M_n\), which destabilizes to a genus \(n\) Heegaard splitting \(V_n\cup_{S_n}W_n\). Combining both results, the author shows that the number of simple closed curves of intersection between \(T_n\) and \(S_n\) is at least \(n\). This gives the first known examples of such lower bounds that are greater than 1.
MSC:
57M99 General low-dimensional topology
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N10 Topology of general \(3\)-manifolds (MSC2010)
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