Combinatorics of CAT(0) cubical complexes, crossing complexes and co-skeletons

Given a polytopal complex \(X\), define its \(k\)th co-skeleton to be the collection of faces of dimension greater than \(k\), i.e. the complement of the \(k\)-skeleton. We study some topological aspects of co-skeletons, in particular deriving a long exact sequence that relates the homologies of co-skeletons to the homologies of links. We will apply these ideas to \(\mathrm{CAT}(0)\) cubical complexes, and conclude that a \(\mathrm{CAT}(0)\) cubical complex has some topological properties (e.g. Cohen–Macaulayness) if and only if a certain associated simplicial complex, the crossing complex, shares these properties.