The Unstabilized Amalgamation of Heegaard Splittings along Disconnected Surfaces

Abstract:

      Let $M$ be a 3-manifold, $\mathcal{F}$$= \{F_{1},F_{2},\ldots,F_{n}\}$ be a collection of essential closed surfaces in $M$ (for any $i,j\in \{1,...,n\}$, if $i\neq j$, $F_{i}$ is not parallel to $F_{j}$ and $F_i\cap F_j=\emptyset$) and $\partial_{0}M$ be a collection of components of $\partial M$. Suppose $M- \bigcup_{F_{i} \in \mathcal{F}} F_{i}\times (-1,1)$ contains $k$ components $M_{1},M_{2},\ldots,M_{k}$. If each $M_{i}$ has a Heegaard splitting $V_{i} \bigcup_{S_{i}} W_{i}$ with $d(S_{i}) > 4(g(M_{1})+ \cdots +g(M_{k}))$, then any minimal Heegaard splitting of M relative to $\partial_{0}M$ is obtained by doing amalgamations and self-amalgamations from minimal Heegaard splittings or $\partial$-stabilization of minimal Heegaard splittings of $M_{1},M_{2}, \ldots, M_{k}$.