Heegaard structures of manifolds in the Dehn filling space (Q1972339)

For a \(3\)-manifold \(X\) with boundary, the Heegaard genus \(g(X)\) is the minimal genus of a closed surface in \(X\) which splits \(X\) into two compression bodies. This genus can change drastically when Dehn filling is performed on \(X\). For example, for any knot complement, there is a Dehn filling which recovers the \(3\)-sphere, and hence decreases the Heegaard genus to \(0\). In this paper, the author shows that such examples are misleading, at least for acylindrical \(X\), by proving that for all but finitely many of the manifolds \(M\) resulting from Dehn filling on a fixed boundary torus of \(X\), the genus satisfies \(g(X)-1\leq g(M)\leq g(X)\). This is a consequence of a penetrating study of how the core circle of the filled-in solid torus can be positioned with respect to Heegaard surfaces in \(X\). In particular, for any collection of Heegaard surface of bounded genus for \(X\), the core circle is isotopic into all of the surfaces for all but finitely many fillings. Additional information on the set of fillings for which the genus drops is obtained. In addition to using the notion of thin position due to D. Gabai, and the geometric techniques commonly found in work with Heegaard splittings, the author uses Cerf theory to find isotopies of families of Heegaard surfaces in \(X\) that move them to surfaces that meet essentially. The application of Cerf theory in this context originated with \textit{H. Rubinstein} and \textit{M. Scharlemann} [Topology 35, No. 4, 1005-1026 (1996; Zbl 0858.57020)].