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**Heegaard structures of negatively curved 3-manifolds.**
*(English)*
Zbl 0890.57025

Let \(M\) be an orientable complete hyperbolic 3-manifold of finite volume and with \(d\) ordered cusps \(c_1,\dots,c_d\). Denote by \(\mathbb{M}\) the collection of all manifolds obtained by Dehn surgery on \(M\), i.e., if \(q\in \mathbb{Q}^d\), \(q=(q_1,\dots,q_d)\), let \(M_q\) denote the manifold obtained by \(q_i\) surgery on the \(i\)-th cusp, \(1\leq i\leq d\), with respect to some prechosen basis for all \(q\). By a Heegaard splitting for \(M\) the authors mean a decomposition of \(M\) into compression bodies as in [A. J. Casson and C. McA. Gordon, Topology Appl. 27, 275-283 (1987; Zbl 0632.57010)]. The main theorem in the paper is

(a) There is a \(d\)-tuple of integers \((N_1,\dots, N_d)\) defining a sub-collection \(\mathbb{M}'\) of \(\mathbb{M}\) by \(\mathbb{M}'= \{M_q\in \mathbb{M}\mid\max(|p_i|,|r_i|) > N_i\), \(i=1,\dots,d\}\) and a finite collection of surfaces, \(\Sigma^1,\dots,\Sigma^r\) embedded in \(M\) so that every irreducible Heegaard splitting surface of genus less than or equal to \(g\) of manifolds \(M_q\in\mathbb{M}'\) is isotopic to one of the \(\Sigma_i\).

(b) A surface \(\Sigma_i\) is either a Heegaard splitting for \(M\) or there is at least one cusp and a unique simple closed curve \(\beta\) on the boundary of a regular neighborhood of the cusp which is isotopic to a simple closed curve \(\beta'\) on \(\Sigma_i\). In this case the curve \(\beta'\) is not isotopic to a core curve of any of the handlebodies determined by \(\Sigma_i\).

(a) There is a \(d\)-tuple of integers \((N_1,\dots, N_d)\) defining a sub-collection \(\mathbb{M}'\) of \(\mathbb{M}\) by \(\mathbb{M}'= \{M_q\in \mathbb{M}\mid\max(|p_i|,|r_i|) > N_i\), \(i=1,\dots,d\}\) and a finite collection of surfaces, \(\Sigma^1,\dots,\Sigma^r\) embedded in \(M\) so that every irreducible Heegaard splitting surface of genus less than or equal to \(g\) of manifolds \(M_q\in\mathbb{M}'\) is isotopic to one of the \(\Sigma_i\).

(b) A surface \(\Sigma_i\) is either a Heegaard splitting for \(M\) or there is at least one cusp and a unique simple closed curve \(\beta\) on the boundary of a regular neighborhood of the cusp which is isotopic to a simple closed curve \(\beta'\) on \(\Sigma_i\). In this case the curve \(\beta'\) is not isotopic to a core curve of any of the handlebodies determined by \(\Sigma_i\).

Reviewer: He Baihe (Changchun)

### MSC:

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

57M50 | General geometric structures on low-dimensional manifolds |

57R65 | Surgery and handlebodies |