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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.

The assertion is false. Take $m=3$ and $n=2$. For the constant attaching map, the cell attachment gives $S^2\vee S^4$. Its degree-two generator has square zero: restricting the square to either sphere gives zero, and restrictions identify its degree-four cohomology with that of the $S^4$ summand.

For the Hopf fibration $\eta :S^3\to S^2={\mathbb{CP}}^1$, the attachment instead gives ${\mathbb{CP}}^2$. One can verify the attaching map explicitly with the characteristic mapIts interior maps homeomorphically to the complement of ${\mathbb{CP}}^1$; on the boundary it sends $(z_1,z_2)$ to $[z_1:z_2:0]$, exactly the Hopf fibration. By part (a), the degree-two generator of $H^{\ast }({\mathbb{CP}}^2;\mathbb{Z})$ has nonzero square generating degree four. HenceThe additive groups agree, but multiplication distinguishes the attachments.

The dimension condition explains why this example is the relevant one. In general the only positive-degree additive generators lie in degrees $n$ and $m+1$. A potentially nonzero product can only be the square of the degree-$n$ generator, and only when $m+1=2n$. Its coefficient is the Hopf invariant of the attaching map. Here the constant map has invariant zero, whereas the complex Hopf fibration has invariant one.