Morse chain complex
= Morse chain complex
{c}
{title2=$C_i=\mathbb Z^{c_i}$}
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>.