The Morse complex of the two-sphere

In polar coordinates $f=\cos \theta$ and the round downward gradient is $\sin \theta {\mathrm{\partial }}_{\theta }$. Multiply it by a smooth positive function of $f$ equal to $4\mathrm{/}(1+f)$ near $N$ and $4\mathrm{/}(1-f)$ near $S$, using cutoffs from [F3] and the positive constant $2$ elsewhere. In the Cartesian radial Morse coordinates of radius $\sqrt{1-f}$ at $N$ and $\sqrt{1+f}$ at $S$, the resulting field is $2w$ and $-2w$, respectively; explicitly they are $w=(x,y)\mathrm{/}\sqrt{1+z}$ near $N$ and $w=(x,y)\mathrm{/}\sqrt{1-z}$ near $S$, hence smooth local Cartesian coordinates with nonsingular derivative at the pole. Thus $X$ satisfies the normalized local model and strictly decreases $f$ elsewhere. The only critical points are $N,S$, with Hessians negative and positive definite and hence indices $2,0$. The unstable set of $N$ and stable set of $S$ are the complementary-pole open disks; the other two sets are single points, so all nonempty stable--unstable intersections are transverse. The smooth field is complete by [F3], so the pair is Morse--Smale. The chain groups in degree one are free on the empty set and are zero.

The differentials out of and into degree one vanish identically: ${\mathrm{\partial }}_2:C{M}_2\to C{M}_1$ has zero target and ${\mathrm{\partial }}_1:C{M}_1\to C{M}_0$ has zero source. The remaining differentials ${\mathrm{\partial }}_0$ and ${\mathrm{\partial }}_3$ have zero target and zero source respectively. So all differentials are zero.

Although every meridian from $N$ to $S$ is a connecting trajectory, its index drop is two. The differential definition in [F2] counts index drop one only, so these trajectories supply no coefficient.

With zero differentials and one generator in degree $2$ and one in degree $0$, the mod-two complex has homology $\mathbb{Z}\mathrm{/}2$ in degrees $0$ and $2$ and zero elsewhere, and the integral complex has homology $\mathbb{Z}$ in degrees $0$ and $2$ and zero elsewhere.