Prism Complexes

Abstract

A prism is the product space ∆ × I where ∆ 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 ber spaces. Using this, we show that a compact 3-manifold admits a special prism complex structure if and only if it is a Seifert ber space with nonempty boundary, a Seifert ber space with a non-empty collection of surfaces in its exceptional set, or a closed Seifert ber space with Euler number zero. So, in particular, a compact 3-manifold with boundary is a Seifert ber space if and only if it has a special prism complex structure.