Morse-Smale complex ($\partial p={\sum }_{\mathrm{ind}q=\mathrm{ind}p-1}n(p,q)q$)

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.

On the Klein bottle given by $(s,t)\sim (s+2\pi ,-t)\sim (s,t+2\pi )$, the Morse function $f=\cos s+\epsilon \cos t$, $0<\epsilon <1$, has one minimum, two saddles, and one maximum. In its descending Morse-Smale gradient flow, the two branches from either saddle to the minimum cancel. From the maximum to the fiber saddle, reflection of the fiber under the base identification reverses the transverse orientation, making the two contributions add. The contributions to the base saddle cancel. With a choice of generator orientations the displayed Morse chain complex follows. Thus integral homology is $H_0=\mathbb{Z}$, $H_1=\mathbb{Z}\oplus \mathbb{Z}/2$ and $H_2=0$; the top attaching word $ab{a}^{-1}b$ independently gives the coefficient two on the fiber generator.

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.