Orientations on simplicial complexes and cubical complexes.

by Masahiro Hachimori


In this paper we discuss orientations of the facet-ridge incidence graphs of regular CW complexes, especially the cases of simplicial complexes and cubical complexes. When the orientation is acyclic and each ridge has in-degree $\ge 1$, it gives a covering by a family of sets associated to each facets. We give a condition when the covering becomes a partition. For simplicial complexes it provides a characterization of shellability, and for cubical complexes we derive inequalities for the Betti numbers of the complexes. Also some additional examples are supplied.