On distance degenerating handlebody fillings
The distance $d(S)$ of a Heegaard splitting $M=V \cup_S W$ of a compact $3$-manifold $M$ into compression bodies $V$ and $W$ is defined as the distance $d_{S}(\mathcal{D}_{S}(V),\mathcal{D}_{S}(W))$ between the set $\mathcal{D}_{S}(V)$ of simple curves on $S$ bounding essential discs in $V$ and the set $\mathcal{D}_{S}(W)$ of simple curves on $S$ bounding essential discs in $W$. Distance here is in the curve complex $\mathcal{C}(S)$ where each vertex corresponds to an isotopy class of a simple closed curve. \par The effect on $d(S)$ of a handlebody filling of $M$ is examined by the author in the following case: There is a single simple curve on $S$ bounding a disc $D_V$ separating $V$ into $V_1$ and $V_2$. $\partial_{+}V_{1} \cap S = S_1$ and $S_1 \cup D_V$ is homeomorphic via a map $f$ to $F_1 = \partial_{-}V_1$. The filling handlebody $H$ has $\partial H = F_1$ so if we set $V \cup_{F_1} H = V_H$ then $V_H \cup_{S_H} W$ is the resulting handlebody filling. \par By defining $\psi$ as the composition of maps \[ \pi_{S}:\mathcal{C}(S) \to \mathcal{C}(S_1)\quad \iota: \mathcal{C}(S_1) \to \mathcal{C}(S_1 \cup D_V)\quad f:\mathcal{C}(S_1 \cup D_V) \to \mathcal{C}(F_1) \] it becomes possible to define the distance $d_{\mathcal{C}(F_1)}(\mathcal{D}(H),\psi(\mathcal{D}(W)))$. Notice $d(S_H) < d(S) = n$. \par The author shows that there exists an $\mathcal{M}$ such that either if $0 < k < \mathcal{M}/2$ and $2k-2 \leq d_{\mathcal{C}(F_1)}(\mathcal{D}(H),\psi(\mathcal{D}(W))) \leq 2k-1$ then $k = d(S_H)$ or $\mathcal{M}/2 < d(S_H) < n$.
scientific article; zbMATH DE number 7047504
scientific article