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

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