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A classical result in Morse theory is the determination of the homotopy type of the loop space of a manifold. In this paper, we study this result through the lens of discrete Morse theory. This requires a suitable simplicial model for the loop space. Here, we use Milnor's $\textrm{F}^+\textrm{K}$ construction to model the loop space of the sphere $S^2$, describe a discrete gradient on it, and identify a collection of critical cells. We also compute the action of the boundary operator in the Morse complex on these critical cells, showing that they are potential homology generators. A careful analysis allows us to recover the calculation of the first homology of $\Omega S^2$., Comment: 15 pages, 1 figure

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 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.