Discrete Morse theory for open complexes

We develop a discrete Morse theory for open simplicial complexes $K=X\setminus T$ 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 $S_K$ of $K$, and the topology change determined by $f$ on $K$ can be understood by analyzing the topology change determined by the discrete Morse function on $S_K$. This topology change is given by a structure theorem on the level subcomplexes of $S_K$. 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. The gradient vector field on $K$ provides a novel alternative to compute Borel-Moore homology.