Discrete Morse theory for open complexes

We develop a discrete Morse theory for open simplicial complexes K=X T where X is a simplicial complex and T a subcomplex of X. A discrete Morse function f on K gives rise to a discrete Morse function on the order complex SK of K, and the topology change determined by f on K can be understood by analyzing the the topology change determined by the discrete Morse function on SK. This topology change is given by a structure theorem on the level subcomplexes of SK. Finally, we show that the Borel-Moore homology of K, a homology theory for locally compact spaces, is isomorphic to the homology induced by a gradient vector field on K and deduce corresponding weak Morse inequalities.