Large-scale homology computations are often rendered tractable by discrete Morse
theory. Every discrete Morse function on a given cell complex X produces a Morse chain
complex whose chain groups are spanned by critical cells and whose homology is isomorphic to
that of X. However, the space-level information is typically lost because very little is known
about how critical cells are attached to each other. In this talk, we discretize a beautiful
construction of Cohen, Jones and Segal in order to completely recover the homotopy type of X
from an overlaid discrete Morse function.