 Morse chain complex

The free chain group in degree $i$ has one oriented generator for each index-$i$ critical point. A cellular chain complex obtained from the Morse handle-attachment theorem computes the integral homology. Under the Morse-Smale gradient flow condition its differential can instead be obtained by signed counts of connecting trajectories. Over a field, if $r_i$ is the rank of ${\partial }_i$, then $c_i=b_i+r_i+r_{i+1}$. This identity implies both weak and strong Morse inequalities.