Prism Complexes
Keywords:
cube complexes, Seifert fiber spaceAbstract
A prism is the product space $\Delta \times I$ where $\Delta$ is a 2 simplex and $I$ is a closed interval. We introduce prism complexes as an analogue of simplicial 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 criterion 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 nonempty 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.
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