Some remarks on Morse theory for posets, homological Morse theory and finite manifolds

We introduce a version of discrete Morse theory for posets. This theory studies the topology of the order complexes K(X) of h-regular posets X from the critical points of admissible matchings on X. Our approach is related to R. Formanʼs discrete Morse theory for CW-complexes and generalizes Forman and Chariʼs results on the face posets of regular CW-complexes. We also introduce a homological variant of the theory that can be used to study the topology of triangulable homology manifolds by means of their order triangulations.