 Morse-Smale complex

Here $n(p,q)$ is the signed number of unparametrized trajectories of a Morse-Smale gradient flow from $p$ to $q$. Compactifying one-dimensional trajectory moduli spaces gives ${\partial }^2=0$. The generators coincide with the cells furnished by the unstable manifolds, and the trajectory count is their cellular incidence number, so the complex computes integral homology.