论文标题
棱镜络合物
Prism complexes
论文作者
论文摘要
棱镜是产品空间$δ\ times i $，其中$δ$是2个简单的，而$ i $是封闭的间隔。作为简单络合物的类似物，我们介绍了棱镜络合物，并表明每个紧凑型$ 3 $ - manifold都有一个棱镜复杂的结构。如果每个内部水平边缘位于四个棱镜中，我们称之为特殊的棱镜复合体，则每个边界水平边缘都位于两个棱镜中，没有水平面位于边界上。我们给出了（可能是不可定向的）Seifert纤维空间中存在水平表面的标准。使用此情况，我们表明，当且仅当它是具有非空边界的Seifert纤维空间时，一个紧凑的3个manifold接受了特殊的Prism复合体结构，这是一个具有非空表面的Seifert纤维空间，其特殊集合或封闭的Seifert纤维空间，具有Euler编号为零。因此，特别是，当它具有特殊的Prism复杂结构时，带有边界的紧凑型$ 3 $ manifold是Seifert光纤空间。
A prism is the product space $Δ\times I$ where $Δ$ is a 2-simplex and $I$ is a closed interval. As an analogue of simplicial complexes, we introduce prism complexes and show that every compact $3$-manifold has a prism complex structure. We call a prism complex special if each interior horizontal edge lies in four prisms, each boundary horizontal edge lies in two prisms and no horizontal face lies on the boundary. We give a criteria for existence of horizontal surfaces in (possibly non-orientable) Seifert fiber spaces. Using this we show that a compact 3-manifold admits a special prism complex structure if and only if it is a Seifert fiber space with non-empty boundary, a Seifert fiber space with a non-empty collection of surfaces in its exceptional set or a closed Seifert fiber space with Euler number zero. So in particular, a compact $3$-manifold with boundary is a Seifert fiber space if and only if it has a special prism complex structure.
