Degenerating slopes with respect to Heegaard distance

Abstract

Let $M=H_{+}{\cup }_S{H}_{-}$ be a genus $g$ Heegaard splitting with Heegaard distance $n\ge \kappa +2$: (1) Let $c_1$, $c_2$ be two slopes in the same component of ${\partial }_{-}{H}_{-}$, such that the natural Heegaard splitting $M^i=H_{+}{\cup }_S(H_{-}{\cup }_{c_i}2-handle)$ has distance less than $n$, then the distance of $c_1$ and $c_2$ in the curve complex of ${\partial }_{-}{H}_{-}$ is at most $3\mathfrak{M}+2$, where $\kappa$ and $\mathfrak{M}$ are constants due to Masur-Minsky. (2) Let $M^{\ast }$ be the manifold obtained by attaching a collection of handlebodies $\mathcal{H}$ to ${\partial }_{-}{H}_{-}$ along a map $f$ from $\partial \mathcal{H}$ to ${\partial }_{-}{H}_{-}$. If $f$ is a sufficiently large power of a generic pseudo-Anosov map, then the distance of the Heegaard splitting $M^{\ast }=H_{+}\cup (H_{-}{\cup }_f\mathcal{H})$ is still $n$. The proofs rely essentially on Masur-Minsky's theory of curve complex.

V2; Added references and corrected typos