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.