= Projective quadratic Morse complex
{title2=$\partial_i=1+(-1)^i$}
For distinct ordered $a_0<\cdots<a_m$, the quotient $\sum a_jx_j^2/\sum x_j^2$ on <Real projective space> has one <critical point> $[e_i]$ of index $i$. Its descending flow is $[x_j(t)]=[e^{-2a_jt}x_j(0)]$. The unstable cell at $[e_i]$ is $\mathbb{RP}^i\setminus\mathbb{RP}^{i-1}$. Between adjacent <critical points> there are two trajectories, whose relative orientation is the degree $(-1)^i$ of the <antipodal map> on $S^{i-1}$. Thus, up to generator orientation, the differential is two in even degrees and zero in odd degrees. This gives the integral <cellular homology of real projective space> rather than only its mod-two ranks.
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