Prism complexes

A prism is the product space $\mathrm{\Delta }\times I$ where $\mathrm{\Delta }$ 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.