具有有限组不相交压缩圆片的Heegaard分解

设$Vcup_SW$是一个闭的三维流形亏格为$g$的, 弱可约的Heegaard分解, 并且在合痕意义下只有有限组位于曲面不同侧的不相交的压缩圆片, 则它存在一个广义的Heegaard分解: $Vcup_SW=(V_1cup_{S_1}W_1)cup_F(W_2cup_{S_2}V_2)$, 并且满足对于每个$i=1,2$, 压缩体$W_i$都只有一个分离的压缩圆片且$d(S_i)geq 2$. 进一步的, 如果有有限且多于1组不相交的压缩圆片, 则至少一个$d(S_i)$等于2, 并且Heegaard曲面满足临界性质.

On Heegaard Splittings with Finitely Many Pairs of Disjoint Compression Disks

Suppose $Vcup_S W$ is a genus-$g$ weakly reducible Heegaard splitting of a closed 3-manifold with finitely many pairs of disjoint compression disks on distinct sides up to isotopy and $g>2$. We show $Vcup_S W$ admits an untelescoping: $(V_1cup_{S_1}W_1)cup_F(W_2cup_{S_2}V_2)$ such that $W_i$ has a unique separating compressing disk and $d(S_i)geq 2$, for $i=1,2$. If there exist more than one but finitely many pairs of disjoint compression disks, at least one of $d(S_i)$ is 2 and $S$ is a critical Heegaard surface.