Stabilizing Heegaard splittings of toroidal 3-manifolds (Q881454)

Let \(V\cup_S W\) be a Heegaard splitting of a closed, orientable \(3\)-manifold \(M\). The Heegaard splitting \(V'\cup_{S'} W'\) is a stabilization of \(V\cup_S W\) if \(W'\) is obtained from \(W\) by attaching a \(1\)-handle \(X\) to \(W\) with core a properly embedded boundary parallel arc in \(V\), and \(V' = \overline{V-X}\), \(S' = V' \cap W'\). A Heegaard splitting \(V\cup_S W\) is an amalgamation along a separating incompressible surface \(F\) in \(M\) if \(S\) can be obtained from \(F\) by a sequence of ambient \(1\)-surgeries on piecewise disjoint arcs embedded in \(M\backslash F\). The author shows that if \(T\) is a separating incompressible torus in \(M\) and the Heegaard splitting \(V\cup_S W\) can be isotoped so that \(V \cap T\) consists of \(k\) annuli, then after at most \(k\) stabilizations, \(V\cup_S W\) is isotopic to an amalgamation along \(T\). In particular, if \(M\) is irreducible and \(T\) is a canonical torus in the JSJ decomposition of \(M\), at most \(4g-4\) stabilizations suffice, where \(g\) is the genus of \(S\). As a Corollary the author obtains an upper bound for the number of stabilizations required so that Heegaard splittings \(P\cup_R Q\) and \(V\cup_S W\) are isotopic, where \(P\cup_R Q\) is obtained from \(V\cup_S W\) by a Dehn twist along \(T\).