Morse Trajectory Moduli Spaces and the Morse Differential

First let the string be $a\to c\to b$, with $k=\lambda (c)$. On $f=f(c)+\epsilon$, write the incoming crossing as $(0,z_{+})$. The slice of $W^u(a)$ is transverse to the stable sphere $S^{+}=\{u=0,\mathrm{\mid }z{\mathrm{\mid }}^2=\epsilon \}$ by [F4]. Its projection to $u\in {\mathbb{R}}^k$ is therefore a submersion at $z_{+}$. By [F5], fixing its surplus local coordinates gives a $k$-disk $D$ of the form $(u,h(u))$, where $h(0)=z_{+}$ and $\mathrm{\mid }h(u){\mathrm{\mid }}^2=\epsilon +\mathrm{\mid }u{\mathrm{\mid }}^2$. Put $g(u)=h(u)\mathrm{/}\mathrm{\mid }h(u)\mathrm{\mid }$. The crossing map to $f=f(c)-\epsilon$, obtained by solving the flow in [F3], sends $(u,h(u))$ to $(\mathrm{\mid }h(u)\mathrm{\mid }u\mathrm{/}\mathrm{\mid }u\mathrm{\mid },\mathrm{\mid }u\mathrm{\mid }g(u))$ for $u\mathrm{\ne }0$. In polar coordinates it extends smoothly to $H(\rho ,\theta )=(\sqrt{\epsilon +{\rho }^2}\theta ,\rho g(\rho \theta )),\rho \ge 0,\theta \in S^{k-1}\mathrm{.}$ At $\rho =0$ its radial derivative has nonzero stable part $g(0)$, independent of the angular derivatives; it is an embedding near each boundary point, with $\rho =\mathrm{\mid }z\mathrm{\mid }$ and $\theta =u\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ recovering its parameters. Its boundary is the unstable sphere $S^{-}$.

At the outgoing crossing $(u_{-},0)$ of $c\to b$, the slice of $W^s(b)$ has codimension $\lambda (b)$ in the exit level and is transverse to $S^{-}$ by [F4]. Local defining equations composed with $H$ thus have a surjective derivative in the angular variables at $(0,u_{-}\mathrm{/}\sqrt{\epsilon })$. Choose an invertible block of $\lambda (b)$ angular coordinates and fix the others. By [F5] they can be solved as smooth functions of $\rho$ for all sufficiently small $\rho \ge 0$, giving points $H(\rho ,\theta (\rho ))\in W^s(b)$. If $\lambda (b)=0$ there are no equations to solve. For $\rho >0$ these points come from $D\subset W^u(a)$ and hence give ordinary trajectories from $a$ to $b$. Their entry and exit points tend to $(0,z_{+})$ and $(u_{-},0)$; smooth finite-time flow dependence controls all other prescribed transversals. They therefore approach the given once-broken trajectory in [F2]. This proves once-broken gluing existence without asserting invertibility of a sliced Fredholm operator.

Induct on the number $r$ of components. For $r=1$, $v$ itself is ordinary. For $r>1$, approximate its first $r-1$ components by an ordinary trajectory $u$ so closely that $u\mathrm{\#}{v}_r$ lies in a prescribed open neighbourhood $W$ of $v$; this is possible by the induction hypothesis and the finite transversal description in [F2]. Since $W$ is open and contains $u\mathrm{\#}{v}_r$, step 2.1 gives an ordinary trajectory in $W$. Hence every neighbourhood of every finite string meets the ordinary stratum, proving density.

If $p,q$ lie in different components, both trajectory spaces are empty. Otherwise work on their compact connected component and fix the compatible metric of [F5] and the auxiliary metric of [F8]. Suppose $f(p)>f(q)$. Each height curve $h_v$ is continuous by its component endpoint limits and [F1]. In a Morse chart of value $\alpha =f(c)$ its Euclidean speed at a noncritical height $a$ satisfies $\mid \frac{d{h}_v}{da}\mid =\frac{1}{2\sqrt{\mathrm{\mid }u{\mathrm{\mid }}^2+\mathrm{\mid }z{\mathrm{\mid }}^2}}\le \frac{1}{2\sqrt{\mathrm{\mid }a-\alpha \mathrm{\mid }}}\mathrm{.}$ Away from finitely many smaller critical charts, $\mathrm{\mid }X\mathrm{\mid }\mathrm{/}\mathrm{\mid }df(X)\mathrm{\mid }$ is bounded on a compact set. In the auxiliary Riemannian metric, [F8] bounds chart norm comparisons on compact chart closures. Thus the length over any height interval of length $\delta$ is at most $L\delta +C{\sum }_{\alpha }\sqrt{\delta }$, uniformly in $v$: integrate $\mathrm{\mid }a-\alpha {\mathrm{\mid }}^{-1\mathrm{/}2}$, whose integral over such an interval is at most $4\sqrt{\delta }$, and add the finitely many chart bounds. The estimate also holds across breaks by continuity and addition of lengths. Since Riemannian distance is bounded by these lengths, it proves equicontinuity in that distance; limits at finitely many breaks preserve the distance bound by continuity. Passage to the compatible metric uses uniform continuity on the compact component.

Let $h_{v_n}\to h$ uniformly. Then $h(f(q))=q$, $h(f(p))=p$ and $f(h(a))=a$. The set $Z$ of heights at which $h(a)$ is critical is finite by [F4] and includes both endpoints. On each component $J$ of $[f(q),f(p)]\setminus Z$, a small compact subinterval has image separated from the critical set. For large $n$ the approximating curves have no break there and solve $h_{v_n}^{\mathrm{\prime }}=X(h_{v_n})\mathrm{/}df(X)(h_{v_n})$. Passing to the integral equation on this compact subinterval gives the same equation for $h$; uniqueness makes its pieces one orbit on all of $J$. Its endpoints are critical by continuity. Reparametrizing by flow time gives a nonconstant full trajectory between them: a smooth flow cannot reach or leave an equilibrium in finite time, by uniqueness in [F3]. Thus every interval between consecutive heights in $Z$ is a connecting component, including any newly acquired break; $h$ is the height map of a broken trajectory. Its length is bounded by [F1]. This proves that the height image is closed in the uniform metric mapping space.

For an ordinary sequence, uniform height convergence to $h_v$ implies shift convergence: select a noncritical height in each component of $v$, translate each trajectory to that crossing, and use smooth flow dependence at the converging initial points in [F3]. Conversely, shift convergence implies pointwise height convergence at every noncritical point of every component, by the unique transverse level crossing. Those heights are dense in the closed height interval. Equicontinuity from step 1.1 and continuity of $h_v$ upgrade convergence on this dense set to uniform convergence: a finite sufficiently fine height mesh controls every value by the triangle inequality. Consequently the two convergence criteria agree. For $r=1$, the same argument identifies the height topology with the compact-open time-translation quotient and its regular-level slice.

By step 1.1 and [F5] the closure of the height image is compact; by step 2.1 it is the image itself. The height metric in [F2] therefore makes $\bar{\mathcal{M}}(p,q)$ compact and metrizable, and [F6] makes it second countable. The image statement is exactly the defining identification in [F2]. If $f(p)\le f(q)$ the spaces are empty by strict descent and the same conclusions hold for the empty image.

The ordinary stratum is open. Otherwise a sequence of broken height maps would approach an ordinary height map; by finiteness in [F4], after a subsequence one of their intermediate critical points would be a fixed $c$, forcing $h_v(f(c))=c$. An ordinary connecting orbit has no intermediate critical point by flow uniqueness, a contradiction. Density is [F7]. Sequential compactness from step 3.1 and [F6], together with step 2.2, proves the asserted subsequence convergence, and [F1] bounds its length.

The components of the given broken trajectory are nonconstant and run from $p_{i-1}$ to $p_i$, so [F1] applies to the string $p=p_0\to p_1\to \cdots \to p_r=q$ and gives $\lambda (p_0)>\lambda (p_1)>\cdots >\lambda (p_r)$; each difference $\lambda (p_{i-1})-\lambda (p_i)$ is a positive integer by [F5], hence at least one.

Telescoping the $r$ drops gives $\lambda (p)-\lambda (q)={\sum }_{i=1}^r(\lambda (p_{i-1})-\lambda (p_i))\ge r$, hence $r\le \lambda (p)-\lambda (q)$.

Each broken trajectory determines its length $r$, its string of critical points $p=p_0,\dots ,p_r=q$ and its tuple $({\gamma }_1,\dots ,{\gamma }_r)\in \mathcal{M}(p_0,p_1)\times \cdots \times \mathcal{M}(p_{r-1},p_r)$, and no two different data give the same broken trajectory by [F4]; conversely a tuple whose string satisfies $\lambda (p_{i-1})>\lambda (p_i)$ for every $i$ yields a broken trajectory, because the components are then nonconstant. Therefore $\bar{\mathcal{M}}(p,q)$ is the disjoint union of the products over $r\ge 1$ and over such strings. Only strictly index-decreasing strings are included, so consecutive points are distinct and every moduli-space factor is defined.

If $\lambda (p)-\lambda (q)=2$ and a broken trajectory has length $r\ge 2$, then step 2.1 gives $r\le 2$, so $r=2$: the trajectory is once-broken and has exactly one intermediate critical point $p_1$. Its two drops are positive integers with sum $2$ by step 1.1, hence both equal $1$, that is $\lambda (p_1)=\lambda (p)-1$.

The union is finite: by step 2.1 only the integers $1\le r\le \lambda (p)-\lambda (q)$ occur (and $\bar{\mathcal{M}}(p,q)=\mathrm{\varnothing }$ when $\lambda (p)\le \lambda (q)$), and for each such $r$ the string is a finite sequence of critical points chosen from the finite set $\mathrm{Crit}(f)$ by [F2]; hence finitely many products occur, each contributing as a single term of the disjoint union irrespective of its cardinality.

Let $(v_n)$ be a sequence in $\mathcal{M}(p,q)$. By [F2] a subsequence converges geometrically to a broken trajectory whose number of components is at most $1$; a broken trajectory has length at least one, so the limit is an ordinary trajectory, that is, an element of $\mathcal{M}(p,q)$. Hence every sequence in $\mathcal{M}(p,q)$ has a convergent subsequence: the space is sequentially compact.

By [F1] the space $\mathcal{M}(p,q)$ is a topological manifold, hence metrizable by [F3]; for metrizable spaces sequential compactness is equivalent to compactness, so step 1.1 makes $\mathcal{M}(p,q)$ compact. This is where the choice principles enter: [A1] supplies the full AC assumed by the metrization result, and $\mathrm{D}\mathrm{C}$ and ${\mathrm{A}\mathrm{C}}_{\omega }$ assumed by the metric compactness equivalence in [F3].

By [F1] the compact space $\mathcal{M}(p,q)$ is discrete, so by [F4] it is finite. For the second clause: whenever a differential on this page is defined by counting trajectories between critical points of Morse index drop one, each of its coefficients is the cardinality (modulo two) or the signed count of a space of the form $\mathcal{M}(p,q)$ with $\lambda (p)-\lambda (q)=1$, and each such space is finite by the first clause; the sums defining the differential therefore have only finitely many nonzero terms, and the total number of coefficients is finite because the critical set of a Morse function on a closed manifold is finite by [F5].

Put $k=\lambda (r)$ and choose entry and exit levels $f(r)\pm \epsilon$. The incoming slice of $W^u(p)$ has dimension $k$ and is transverse to the stable sphere $S^{+}$ at its crossing $(0,z_{+})$. By [F5], a local sheet is a disk $D=\{(u,h(u))\}$ with $h(0)=z_{+}$ and $\mathrm{\mid }h(u){\mathrm{\mid }}^2=\epsilon +\mathrm{\mid }u{\mathrm{\mid }}^2$. The flow passage in [F3], with $g=h\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$, extends in polar coordinates to $H(s,\theta )=(\sqrt{\epsilon +s^2}\theta ,sg(s\theta )),(s,\theta )\in [0,\delta )\times S^{k-1}\mathrm{.}$ This is a smooth embedded collar $Q$ of the unstable sphere $S^{-}$: its radial derivative at $s=0$ has nonzero stable component $g(0)$, the angular derivatives span the sphere tangent space, and $s=\mathrm{\mid }z\mathrm{\mid }$, $\theta =u\mathrm{/}\mathrm{\mid }u\mathrm{\mid }$ recover its parameters. For $s>0$, $Q$ is the passage image of $D\setminus \{(0,z_{+})\}$.

At the outgoing crossing $(u_{-},0)$, the stable slice of $W^s(q)$ has codimension $k-1$ in the exit level and is transverse to the $(k-1)$-sphere $S^{-}$ by [F2]. In local sphere coordinates its defining equations composed with $H$ thus have invertible angular derivative. Apply [F5], extending the formula for $H$ to negative $s$ for this local calculation, to solve uniquely $\theta =\theta (s)$ near $(0,u_{-}\mathrm{/}\sqrt{\epsilon })$. If $k=1$ there are no angular variables or equations and the assertion is immediate. The curve $\chi (s)=H(s,\theta (s))$ is a smooth embedded half-interval in $Q\cap W^s(q)$; for $s>0$ it lies in $W^u(p)$ as well. By [F4] it determines a smooth embedded family of ordinary orbit classes $\psi (s)$. The inverse passage gives entry points tending to $(0,z_{+})$ and exit points tending to $(u_{-},0)$, so [F1] makes the extension $\psi (0)=({\gamma }_1,{\gamma }_2)$ continuous.

Any trajectory sufficiently close to the broken point in the transversal neighbourhoods of [F1] has its entry point in $D$: closeness at the endpoint exit sphere of $p$ and smooth finite-time transport along the incoming component select precisely this local sheet of $W^u(p)$. Its exit point is therefore in $Q$, and closeness at the endpoint entry sphere of $q$ likewise selects the local sheet of $W^s(q)$. The uniqueness in step 2.1 forces that exit point to be $\chi (s)$. Shrinking the neighbourhood gives exactly one such half-interval; its inverse coordinate $s=\mathrm{\mid }z\mathrm{\mid }$ is continuous, including at the broken point. This proves the open-neighbourhood, homeomorphism and eventual-containment claims. Finally the passage time is $\frac{1}{2}\log (\sqrt{\epsilon +s^2}\mathrm{/}s)$ by [F3], which diverges as $s\to 0^{+}$.

By [F1] the interior part $\mathcal{M}(p,q)$ is a smooth $1$-manifold, and by [F2] it is open in $\bar{\mathcal{M}}(p,q)$. Its points are interior points of the compactification: the charts of the manifold structure of [F1] are charts of $\bar{\mathcal{M}}(p,q)$ around them.

The added points are the broken trajectories of length at least two, and by [F3] each of them is a once-broken trajectory with intermediate point $r$ of index $\lambda (p)-1$. Conversely every once-broken trajectory with such an $r$ is a broken trajectory of length two and is not ordinary, hence an added point. So the boundary set is exactly ${⨆}_{\lambda (r)=\lambda (p)-1}\mathcal{M}(p,r)\times \mathcal{M}(r,q)$ as a set, and each of its points has a collar chart $[0,\delta )$ by [F5]. There are finitely many added points by [F4], and the metric topology of [F2] allows the collars to be shrunk to pairwise disjoint neighbourhoods. On an overlap with an interior chart, the collar and its inverse are smooth because its interior restriction is a smooth embedding of one-dimensional manifolds. There are no overlaps between different boundary charts after this shrinking. Thus the collars and interior charts give a compatible smooth atlas as required by [F6]; hence $\bar{\mathcal{M}}(p,q)$ is a smooth $1$-manifold with boundary whose interior is $\mathcal{M}(p,q)$ and whose boundary is that set.

The boundary identification is a homeomorphism onto the disjoint union of the products: the factors are discrete spaces (by the index-one finiteness and discreteness in [F4]), finitely many by [F4], and a sequence of once-broken trajectories with a fixed intermediate point $r$ converges geometrically to $({\gamma }_1,{\gamma }_2)$ exactly when its components converge to ${\gamma }_1$ and ${\gamma }_2$ in the geometric topologies, which is the product of the discrete topologies; distinct intermediate points $r$ give disjoint factors because a once-broken trajectory determines its intermediate critical point. Since each factor is a finite discrete space by [F4], the boundary is a finite disjoint union of finite discrete spaces.

Compactness, metrizability and second countability are [F2]; the collar charts of [F5] show that the compactification is obtained from $\mathcal{M}(p,q)$ by adding the once-broken trajectories, completing the proof.

Let $p\in {\mathrm{Crit}}_k(f)$ and $q\in {\mathrm{Crit}}_{k-2}(f)$. Expanding the definition, the coefficient of $q$ in ${\mathrm{\partial }}_{k-1}({\mathrm{\partial }}_{k}p)$ is the sum over $r\in {\mathrm{Crit}}_{k-1}(f)$ of the products $n_2(p,r)n_2(r,q)$ in $\mathbb{Z}\mathrm{/}2$; both index drops here are equal to one, so both factors are parities of finite cardinalities by [F1], and the product of the two parities is the parity of the cardinality of the product $\mathcal{M}(p,r)\times \mathcal{M}(r,q)$. Hence the coefficient equals the parity of the cardinality of the finite disjoint union ${⨆}_{r\in {\mathrm{Crit}}_{k-1}(f)}\mathcal{M}(p,r)\times \mathcal{M}(r,q)$.

Here $\lambda (p)-\lambda (q)=2$, so by [F2] the disjoint union of step 1.1 is exactly the boundary of the compact $1$-manifold with boundary $\bar{\mathcal{M}}(p,q)$; hence the coefficient of $q$ in ${\mathrm{\partial }}^{2}p$ is $\mathrm{\#}\mathrm{\partial }\bar{\mathcal{M}}(p,q)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>2$. For any critical point $q^{\mathrm{\prime }}$ that is not of index $k-2$, the coefficient of $q^{\mathrm{\prime }}$ in ${\mathrm{\partial }}^{2}p$ is zero because ${\mathrm{\partial }}_{k-1}$ takes values in $C{M}_{k-2}(f,X;\mathbb{Z}\mathrm{/}2)$, whose basis is ${\mathrm{Crit}}_{k-2}(f)$.

By [F3] the boundary of the compact $1$-manifold $\bar{\mathcal{M}}(p,q)$ has even cardinality, so the coefficient of every $q$ of index $k-2$ in ${\mathrm{\partial }}^{2}p$ vanishes in $\mathbb{Z}\mathrm{/}2$; by step 2.1 all other coefficients vanish as well. Hence ${\mathrm{\partial }}_{k-1}\circ {\mathrm{\partial }}_k=0$ on basis elements, and therefore on all of $C{M}_k(f,X;\mathbb{Z}\mathrm{/}2)$ by linearity.

Since this holds for every $k$, the pair $(C{M}_{\ast }(f,X;\mathbb{Z}\mathrm{/}2),\mathrm{\partial })$ satisfies the defining condition of a chain complex in the abelian category of $\mathbb{Z}\mathrm{/}2$-modules by [F5]; its homology is the mod-two Morse homology of $(f,X)$ by definition.