沿着不连通曲面的Heegaard分解的不可稳定化的融合积

The Unstabilized Amalgamation of Heegaard Splittings along Disconnected Surfaces

中文关键词:  不可稳定化  距离  融合积  Heegaard分解.

英文关键词:unstabilized  distance  amalgamation  Heeaggard splitting.

中文摘要:

      令$M$为一个三维流形, $\mathcal{F}=\{F_{1},F_{2},\ldots,F_{n}\}$为在$M$中的一族互不相交且互不平行的本质闭曲面,同时$\partial_0M$为$\partial M$的若干边界连通分支.假设$M-\bigcup_{F_{i} \in \mathcal{F}} F_{i}\times (-1,1)$有$k$个分支$M_1,M_2,\ldots,M_k$.若每个$M_i$都有一个Heegaard分解$V_{i} \bigcup_{S_{i}} W_{i}$且$d(S_i)>4(g(M_1)+\cdots+g(M_k))$,则任一$M$相对于$\partial_0M$的最小Heegaard分解都是由$M_1,M_2,\ldots,M_k$的最小Heegaard分解其稳定化做融合积和自融合积获得.

英文摘要:

      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 $M_{1},M_{2}, \ldots, M_{k}$.