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A Morse function is a smooth real-valued function all of whose critical points have nonsingular Hessian matrices. We use the descending convention for a gradient-like vector field $V$: away from critical points, $df(V)<0$, and near a critical point of Morse index $\lambda$, in Morse lemma coordinates $f=f(p)-|u{|}^2+|v{|}^2$, the field can be chosen as $V=(2u,-2v)$.

For its flow ${\varphi }_t$, define the unstable manifold $W^u(p)=\{x:{\lim }_{t\to -\infty }{\varphi }_t(x)=p\}$ and the stable manifold $W^s(q)=\{x:{\lim }_{t\to +\infty }{\varphi }_t(x)=q\}$. Their dimensions are $\mathrm{ind}p$ and $m-\mathrm{ind}q$. The Morse-Smale transversality condition isFor a closed manifold, this lets us form the Morse-Smale complex over $\mathbb{Z}$. Orient each unstable manifold and takeThe transverse intersection for an index difference of one is one-dimensional; quotienting by flow time gives a finite set of trajectories, and $n(p,q)$ counts them with orientation signs. Compactifying the one-dimensional spaces of trajectories for an index difference of two adds broken trajectories. Their oriented boundary count is zero, giving ${\partial }^2=0$. The descending cells are the unstable manifolds, and these counts are the incidence numbers of the corresponding cellular chain complex, so the homology of this complex is $H_{\ast }(M;\mathbb{Z})$. Ambient orientability is unnecessary: orientations of the unstable cells provide the requisite coorientations and trajectory signs.

For the Klein bottle, use the quotient of ${\mathbb{R}}^2$ by $(s,t)\sim (s+2\pi ,-t)$ and $(s,t)\sim (s,t+2\pi )$. A model of the standard four-critical-point height function isIt is well defined under both identifications. Its critical points, their Morse indices, and critical values areThe Hessian matrix is diagonal with entries $-\cos s$ and $-\epsilon \cos t$, so each point is nondegenerate. The flat quotient metric gives descending equations $\dot{s}=\sin s$, $\dot{t}=\epsilon \sin t$. Their saddle separatrices are the coordinate lines; the two saddle-to-saddle intersections are empty, and all remaining stable and unstable intersections are transverse. This is a Morse-Smale gradient flow. If one insists on the exact local-model version of a gradient-like vector field above, change the two coordinate speeds near the critical points to the standard quadratic-model speeds. This preserves the separatrices, signs and transversality.

Both branches of either saddle flow to the minimum with opposite endpoint signs, so $\partial a=\partial b=0$. From the maximum to $a$, the two trajectories lie on $s=0$ and approach $t=\pi$ from opposite directions. The identification in $t$ is a translation, so their incidence signs are opposite and cancel. The two trajectories to $b$ lie on $t=0$ and approach $s=\pi$ from opposite directions. The identification in $s$ reverses the transverse $t$-coordinate, reversing one additional orientation sign. These two contributions consequently have the same sign. Choosing the orientation of $b$ gives $\partial q=2b$.

One can check the coefficient directly from the cell attachment of the top cell: its boundary word is $ab{a}^{-1}b$, with exponent sums zero on the base loop $a$ and two on the fiber loop $b$. Thus the integral Morse complex of the Klein bottle isandCounting trajectories without signs, or working only modulo two, would miss the order-two class.