The Topology of Cellular Gradient Flow

Large-scale homology computations are often rendered tractable by discrete Morse theory. A discrete Morse function on a cell complex X is a recipe for partitioning the cells into two classes: one consists of critical cells (these play the role of singular points of smooth Morse functions) while the other consists of regular cells (these generate gradient trajectories between critical cells). This partition produces a new chain complex whose chain groups are spanned by the critical cells and whose homology is isomorphic to that of X. However, the refined space-level information is typically lost in this process because very little is known about how these critical cells are attached to each other. In this talk, we will discretise a gorgeous construction of R Cohen, J Jones and G Segal in order to completely recover the homotopy type of X from an overlaid discrete Morse function.