Projective quadratic Morse complex (${\partial }_i=1+(-1{)}^i$)

For distinct ordered $a_0<\cdots <a_m$, the quotient $\sum a_j{x}_j^2/\sum x_j^2$ on Real projective space has one critical point $[e_i]$ of index $i$. Its descending flow is $[x_j(t)]=[e^{-2a_{j}t}{x}_j(0)]$. 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.