Lusternik–Schnirelmann category for cell complexes and posets

Abstract

This paper introduces two analogues of the Lusternik–Schnirelmann category from a combinatorial viewpoint. One analogue is defined for finite cell complexes using their subcomplexes and simple homotopy theory; the other is an invariant for finite posets with respect to simple equivalence based on the notion of weak beat points. We examine the relation between these two invariants by taking the face posets of complexes or order complexes of posets.