Lusternik-Schnirelmann category for cell complexes and posets (Q327398)

There are plenty of versions of the Lusternik-Schnirelmann category of a space \(X\): each one is the least integer \(n\) such that \(X\) can be covered by \(n+1\) open subspaces that are either (a) contractible in \(X\) (cat), (b) contractible in themselves (gcat), or (c) weakly contractible (qgcat). This paper sets out a `combinatorial' version of the Lusternik-Schnirelmann category for finite cellular complexes, or rather two versions, to be precise: one, denoted by cgcat (`c' stands for `combinatorial'), corresponds to the original LS-category; the other, denoted by cCat, corresponds to the strong category of Ganea. The author proves that cgcat(\(Y\)) = qgcat(\(\chi(Y)\)) where \(\chi(Y)\) is the face poset of a finite regular cell complex \(Y\). One might expect that, conversely, cgcat(\(K(P)\)) = qgcat(\(P\)) where \(K(P)\) is the order complex of a finite poset \(P\) (considered as a finite \(T_0\)-space). However this is not true in general. Nevertheless, the equality holds in the case of the strong version: cCat(\(K(P)\)) = qCat(\(P\)). Finally, the author gives the discrete version of the classical Lusternik-Schnirelmann theorem about critical points. For what he calls `discrete Morse function' \(f: \chi(Y) \to \mathbb{R}\), assuming that \(f\) is injective, with \(n+1\) `critical cells', he shows that cgcat\((Y) \leq n\).