Directed path spaces via discrete vector fields

Let K be an arbitrary semi-cubical set that can be embedded in a standard cube. Using Discrete Morse Theory, we construct a CW-complex that is homotopy equivalent to the space P→(K)vw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {P}(K)_v^w$$\end{document} of directed paths between two given vertices v, w of K. In many cases, this construction is minimal: the cells of the constructed CW-complex are in 1–1 correspondence with the generators of the homology of P→(K)vw\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\vec {P}(K)_v^w$$\end{document}.