Unstable orientations induce orientations of the trajectory moduli spaces

Fix a critical point $y$, a Morse chart as in [F3] and write its coordinates as $(u,v)$ with $\mathrm{\mid }u\mathrm{\mid }$ of length $\lambda (y)$. Let $E^u$ be the subbundle of $TM$ over $W^s(y)\cap U$ whose fibre at a point $(0,v)$ is the span of the coordinate fields ${\mathrm{\partial }}_{u_1},\dots ,{\mathrm{\partial }}_{u_{\lambda (y)}}$; since $W^s(y)\cap U=\{u=0\}$ in these coordinates, $T_{(0,v)}M=T_{(0,v)}{W}^s(y)\oplus E_{(0,v)}^u$ is a direct sum, and $E^u$ is a smooth rank-$\lambda (y)$ subbundle. In the same coordinates ${\mathrm{\Phi }}_t(u,v)=(e^{2t}u,e^{-2t}v)$, so $D{\mathrm{\Phi }}_t$ carries $E_z^u$ isomorphically onto $E_{{\mathrm{\Phi }}_t(z)}^u$: the local flow preserves the subbundle.

The ray $o{r}_y$ is a ray in $\det E_y^u$ by [F1]. Let $r$ be the ray field on $E^u$ over $W^s(y)\cap U$ that is constant in the coordinates of step 1.1 and equals $o{r}_y$ in the fibre at $y$. Because $D{\mathrm{\Phi }}_t$ acts on that coordinate frame by the positive factor $e^{2t}$, it maps the ray at $z$ to the ray at ${\mathrm{\Phi }}_t(z)$; hence $r$ is a smooth flow-invariant orientation of $E^u$ near $y$, and it co-orients $W^s(y)$ there.

Let $p\in W^s(y)$; then ${\mathrm{\Phi }}_t(p)\to y$ as $t\to \mathrm{\infty }$ by [F2], so for all sufficiently large $t$ the points ${\mathrm{\Phi }}_t(p)$ lie in $W^s(y)\cap U$ and the flow segments joining them stay in $U$, by the explicit local model of [F3]. For any such $t$ put $E_p^u:=D{\mathrm{\Phi }}_{-t}\left(E_{{\mathrm{\Phi }}_t(p)}^u\right)$ and $r_p:=D{\mathrm{\Phi }}_{-t}\left(r_{{\mathrm{\Phi }}_t(p)}\right)$. If $t\le t^{\mathrm{\prime }}$ are both large enough, then flow-invariance within the chart gives $D{\mathrm{\Phi }}_{-(t^{\mathrm{\prime }}-t)}(E_{{\mathrm{\Phi }}_{t^{\mathrm{\prime }}}(p)}^u)=E_{{\mathrm{\Phi }}_t(p)}^u$ and $D{\mathrm{\Phi }}_{-(t^{\mathrm{\prime }}-t)}(r_{{\mathrm{\Phi }}_{t^{\mathrm{\prime }}}(p)})=r_{{\mathrm{\Phi }}_t(p)}$, so $D{\mathrm{\Phi }}_{-t^{\mathrm{\prime }}}=D{\mathrm{\Phi }}_{-t}\circ D{\mathrm{\Phi }}_{-(t^{\mathrm{\prime }}-t)}$ shows that the pair $(E_p^u,r_p)$ does not depend on $t$. Since $D{\mathrm{\Phi }}_{-t}$ is a linear isomorphism it preserves direct sums, so $T_{p}M=T_p{W}^s(y)\oplus E_p^u$ for every $p\in W^s(y)$; the choice of $t$ can be made locally constant in $p$ by continuity of the flow, so $E^u$ is a smooth subbundle of $TM{\mathrm{\mid }}_{W^s(y)}$ and $r$ a smooth ray field on it, i.e. a co-orientation of $W^s(y)$. The complement may depend on the Morse chart, but its orientation induces an orientation of the canonical normal quotient $TM{\mathrm{\mid }}_{W^s(y)}\mathrm{/}T{W}^s(y)$. Any other chart gives a continuous orientation of this same quotient agreeing with it at $y$; the sign comparing them is locally constant on the connected manifold $W^s(y)$, so they agree everywhere. Thus the co-orientation, rather than the complement subbundle, is canonical.

Let $x,y$ be critical points with $W^u(x)⋔W^s(y)$ and let $p\in W^u(x)\cap W^s(y)$. The co-orientation of step 3.1 orients the normal bundle $\nu :=TM{\mathrm{\mid }}_{W^s(y)}\mathrm{/}T{W}^s(y)\cong E^u$ of $W^s(y)$, and the orientation $o{r}_x$ of [F1] orients $T_p{W}^u(x)$. The composition $T_p{W}^u(x)\to T_{p}M\to {\nu }_p$ is onto with kernel $T_p(W^u(x)\cap W^s(y))$, because transversality says $T_p{W}^u(x)+T_p{W}^s(y)=T_{p}M$; thus $0\to T_p(W^u(x)\cap W^s(y))\to T_p{W}^u(x)\to {\nu }_p\to 0$ is exact. By the two-of-three determinant-line rule of [F4], applied through the ordered splitting of [F5], the orientation of the middle term together with the orientation of the quotient determines a ray in the determinant line of the kernel, i.e. an orientation of $T_p(W^u(x)\cap W^s(y))$; both inputs are smooth in $p$, and the pointwise comparison of two local constructions is locally constant by [F7], so these rays form a smooth orientation of $W^u(x)\cap W^s(y)$, canonical in the given data. Under the identification ${\mathrm{ev}}_0$ of [F6] this orients $\tilde{\mathcal{M}}(x,y)$.

Now assume $\lambda (x)-\lambda (y)=1$. By [F6] the identification with the transverse intersection gives $\dim \tilde{\mathcal{M}}(x,y)=\lambda (x)+(n-\lambda (y))-n=1$, and no point of $W^u(x)\cap W^s(y)$ is critical, so the flow vector $X$ is nowhere zero on the one-manifold $\tilde{\mathcal{M}}(x,y)$. Let $L$ be an orbit of the flow. The orbit map $t\mapsto t\cdot \gamma$ is injective by [F9], so $L$ is the injective continuous image of $\mathbb{R}$; it is open in $\tilde{\mathcal{M}}(x,y)$, since near any of its points a flow box for the nowhere-zero field $X$ (existence and uniqueness for the flow, [F2]) exhibits the local orbits as the connected components of a small chart; and it is closed, because a limit point in $\tilde{\mathcal{M}}(x,y)$ of points of $L$ lies on the same local orbit as they do, hence in $L$. A nonempty subset of a manifold that is open, closed and connected is a connected component, so each component of $\tilde{\mathcal{M}}(x,y)$ is exactly one flow line. Its tangent space at any point is the line $\mathbb{R}\cdot X$, and the comparison of the orientation of step 4.1 with the positive flow ray ${\mathbb{R}}_{>0}\cdot X$ is locally constant by [F7]; on the connected component this comparison is therefore a constant sign $ϵ(\gamma )\in \{+1,-1\}$, which is the asserted comparison sign.

It remains to orient the quotient $\mathcal{M}(x,y)$ of dimension $\lambda (x)-\lambda (y)-1$. For a trajectory $\gamma$, choose a regular value $c$ with $f(y)<c<f(x)$; by [F8] the tangent space of the parametrized space splits canonically as $T_{\gamma }\tilde{\mathcal{M}}(x,y)=\mathbb{R}\cdot \dot{\gamma }\oplus T_{\gamma }\mathcal{M}(x,y)$ with $\dot{\gamma }=X(\gamma (0))$ nonzero, and the orientation of the first summand is the positive flow orientation. By the ordered product isomorphism of [F5] the orientation of $T_{\gamma }\tilde{\mathcal{M}}(x,y)$ from step 4.1 determines a unique ray in $\det T_{\gamma }\mathcal{M}(x,y)$, namely the ray whose tensor product with the positive flow ray is the intersection orientation. This ray varies smoothly with $\gamma$ because the splitting and both given orientations do, so it is an orientation of $\mathcal{M}(x,y)$; the flow-first convention names this choice, and it does not depend on the auxiliary regular value $c$, since the quotient tangent space is the same for every level representative.

Finally, replacing $o{r}_x$ by its opposite reverses the ray in $\det T_p{W}^u(x)$ and leaves the co-orientation of $W^s(y)$ unchanged, so the kernel orientation of step 4.1 is reversed by the same determinant-line rule; replacing $o{r}_y$ by its opposite reverses the ray $o{r}_y$ of step 2.1 and hence the co-orientation of step 3.1, and by the same rule reverses the kernel orientation again; in both cases the orientations of steps 5.1 and 5.2, being determined by the intersection orientation, are reversed as asserted.