03.15.07 · modern-geometry / morse-homology

Continuation maps and invariance of $H M_{*}$

shipped3 tiersLean: none

Anchor (Master): Schwarz *Morse Homology* Part II Ch. 5; Salamon 1990; Conley-Zehnder 1984

Intuition Beginner

You have built a counting machine. Feed it a height function on a curvy surface, plus a way of measuring distances, and it returns a small algebraic gadget — a list of resting points sorted by how many downhill directions each one has, wired together by signed counts of the streams between them. The homology of that gadget is meant to be a fact about the surface, not about the particular height function you happened to pour in. The worry is plain: a different height function gives different resting points and different streams, so why should the answer come out the same?

The fix is a bridge. Suppose you have two height functions on the same surface. Tilt one slowly into the other, so that at each moment in between you have a perfectly good height function. Now look at the streams that flow as the function itself is changing — paths that start at a resting point of the first function and end at a resting point of the second. Counting these moving-target streams builds a translator: a rule that turns the gadget of the first function into the gadget of the second.

The real content is that this translator respects all the wiring, and that translating one way and then back is the same as doing nothing — up to a harmless adjustment. That is enough to force the two homologies to agree. The auxiliary choices you made were never seen by the answer.

Visual Beginner

Alt text: Two upright tori with separate height functions, joined by a band that morphs the left function into the right along a time axis labelled s. Sample streams cross the band from a left resting point to a right resting point. Below, a forward arrow and a backward arrow with a small loop marked equal to doing nothing up to adjustment. The picture conveys that a slow morph of one height function into another builds a translator between the two counting gadgets and that translating forward then back changes nothing essential.

Worked example Beginner

Take a round sphere standing upright, with the ordinary height function: one top resting point $N$ (a maximum) and one bottom resting point $S$ (a minimum). The counting gadget has a slot for $N$ , a slot for $S$ , and no streams worth counting between same-level points, so its homology is one copy of the integers at the bottom and one at the top — the homology of the sphere.

Now nudge the sphere: push the top dimple a little to the side, giving a second height function with its own top $N^{'}$ and bottom $S^{'}$ , in slightly shifted positions. Build the bridge by sliding the first shape into the second over a span of time. As the function morphs, exactly one moving-target stream runs from $N$ to $N^{'}$ and one from $S$ to $S^{'}$ . So the translator sends the slot for $N$ to the slot for $N^{'}$ and the slot for $S$ to the slot for $S^{'}$ , each with a plus sign.

Translating the bottom integer copy and the top integer copy across, then sliding back the other way, returns each slot to itself. The count of one stream each way composes to the count of doing nothing.

What this tells us: the translator is a clean match between the two gadgets, so their homologies are the same one copy of the integers at bottom and top. The shift of the dimple, an auxiliary choice, left no trace on the final answer.

Check your understanding Beginner

Formal definition Intermediate+

Fix a closed Riemannian manifold $M^{n}$ and two Morse-Smale pairs $(f_{0}, g_{0})$ and $(f_{1}, g_{1})$ , each with its Morse complex $(C_{*} (f_{α}), \partial_{α})$ as built in 03.15.05 and assembled in 03.15.06: $C_{*} (f_{α}) = ⨁_{x \in Crit (f_{α})} Z ⟨ x ⟩$ graded by the Morse index, with $\partial_{α} x = \sum_{μ (y) = μ (x) - 1} n_{α} (x, y) y$ .

A homotopy of data is a smooth family $(f_{s}, g_{s})_{s \in R}$ of functions and metrics on $M$ that is constant outside a compact $s$ -interval: there is $R > 0$ with $(f_{s}, g_{s}) = (f_{0}, g_{0})$ for $s \leq - R$ and $(f_{s}, g_{s}) = (f_{1}, g_{1})$ for $s \geq R$ . Such a family defines the non-autonomous negative-gradient equation for a path $u : R \to M$ , $$ \frac{du}{ds} + \nabla_{g_s} f_s(u(s)) = 0, $$ where $\nabla_{g_{s}}$ is the gradient with respect to the time- $s$ metric. Because the equation is not invariant under $s$ -translation, its solutions are honest paths, not equivalence classes. For critical points $x \in Crit (f_{0})$ and $y \in Crit (f_{1})$ , write $$ \mathcal{M}^{H}(x, y) ;=; \Big{, u : \mathbb{R} \to M ;\Big|; \tfrac{du}{ds} + \nabla_{g_s} f_s(u) = 0,\ \ u(-\infty) = x,\ u(+\infty) = y ,\Big} $$ for the space of continuation trajectories of the homotopy $H = (f_{s}, g_{s})$ . The asymptotic limits are required in the exponential-convergence sense of 03.15.03.

The linearisation of the section $u \mapsto \frac{d u}{d s} + \nabla_{g_{s}} f_{s} (u)$ along a solution is again an asymptotically-constant first-order operator $D_{u}^{H} = \frac{d}{d s} + A (s)$ , with $A (s) \to Hess_{g_{0}} f_{0} (x)$ as $s \to - \infty$ and $A (s) \to Hess_{g_{1}} f_{1} (y)$ as $s \to + \infty$ . Its Fredholm index is $$ \operatorname{ind} D^H_u ;=; \mu(x) - \mu(y), $$ the spectral flow between the two asymptotic Hessians 03.15.02. For a generic homotopy $H$ the section is transverse, so each $M^{H} (x, y)$ is a smooth manifold of dimension $μ (x) - μ (y)$ . The decisive change from the autonomous case is the absence of the $R$ -quotient: the dimension is $μ (x) - μ (y)$ , not $μ (x) - μ (y) - 1$ , so when $μ (x) = μ (y)$ the moduli is a compact zero-manifold — a finite signed set.

A coherent orientation of these continuation moduli, built exactly as in 03.15.05 from the determinant lines of $D_{u}^{H}$ , assigns each isolated $u \in M^{H} (x, y)$ (the case $μ (x) = μ (y)$ ) a sign $n^{H} (u) \in {\pm 1}$ . The continuation map $Φ^{H} : C_{*} (f_{0}) \to C_{*} (f_{1})$ is the degree- $0$ homomorphism $$ \Phi^H x ;=; \sum_{\substack{y \in \mathrm{Crit}(f_1)\ \mu(y) = \mu(x)}} \Big(\sum_{u \in \mathcal{M}^{H}(x,y)} n^H(u)\Big), y . $$

Counterexamples to common slips

The continuation map is degree $0$ , not degree $- 1$ . The Morse differential $\partial$ counts index-difference-one trajectories of a fixed function (a one-dimensional quotient moduli); $Φ^{H}$ counts index-difference-zero solutions of the moving function (a zero-dimensional non-quotient moduli). The missing $- 1$ is exactly the lost translation symmetry.
$Φ^{H}$ depends on the whole homotopy $H$ , not only on its endpoints $(f_{0}, g_{0})$ and $(f_{1}, g_{1})$ . Two homotopies between the same endpoints can give different chain maps; what the next theorem secures is that they are chain homotopic, hence equal on homology, not equal as maps.
A homotopy need not be monotone in any energy sense for the chain-map identity to hold; the cutoff-outside-a-compact-interval condition and transversality are what is needed. Monotonicity matters for the parallel story of filtration-preserving maps, not for invariance per se.
Reversing the homotopy, $\overset{ˉ}{H}_{s} = H_{- s}$ with endpoints swapped, gives a continuation $Φ^{\overset{ˉ}{H}} : C_{*} (f_{1}) \to C_{*} (f_{0})$ ; it is not the inverse matrix of $Φ^{H}$ , only a chain-homotopy inverse.

Key theorem with proof Intermediate+

The construction earns its name through three identities, proved one after another from the structure of the low-dimensional continuation moduli ^{[Schwarz Part II Ch. 5]}. Together they make $H \mapsto Φ^{H}$ an invariance machine.

Theorem (continuation maps and invariance). Let $(f_{0}, g_{0}), (f_{1}, g_{1})$ be Morse-Smale pairs on a closed Riemannian manifold $M$ .

(Chain map.) For any generic homotopy $H$ from $(f_{0}, g_{0})$ to $(f_{1}, g_{1})$ , the map $Φ^{H}$ satisfies $\partial_{1} Φ^{H} = Φ^{H} \partial_{0}$ .
(Homotopy independence.) If $H$ and $H^{'}$ are two such homotopies, then $Φ^{H}$ and $Φ^{H^{'}}$ are chain homotopic: there is a degree- $+ 1$ map $K$ with $Φ^{H^{'}} - Φ^{H} = \partial_{1} K + K \partial_{0}$ .
(Composition and identity.) The continuation of the constant homotopy is the identity up to chain homotopy, and the continuation of a concatenation $H_{2} * H_{1}$ is chain homotopic to $Φ^{H_{2}} Φ^{H_{1}}$ .

Consequently $Φ^{H}$ induces an isomorphism $(Φ^{H})_{*} : H M_{*} (f_{0}, g_{0}) ≅ H M_{*} (f_{1}, g_{1})$ , independent of the homotopy $H$ , and these isomorphisms compose; in particular $H M_{*} (f_{0}, g_{0}) ≅ H M_{*} (f_{1}, g_{1})$ canonically.

Proof. (1) Chain map. Fix $x \in Crit (f_{0})$ and $w \in Crit (f_{1})$ with $μ (w) = μ (x) - 1$ . The coefficient of $w$ in $(\partial_{1} Φ^{H} - Φ^{H} \partial_{0}) x$ is computed from the one-dimensional moduli $M^{H} (x, w)$ , which has $μ (x) - μ (w) = 1$ , hence dimension $1$ . By the compactness analysis of 03.15.03 adapted to the non-autonomous equation, $M^{H} (x, w)$ is precompact and its boundary in the broken-trajectory compactification consists of two kinds of split configurations: a $\partial_{0}$ -trajectory of $f_{0}$ from $x$ to an intermediate $z \in Crit (f_{0})$ with $μ (z) = μ (w)$ followed by a continuation $z ⇝ w$ (an $f_{0}$ -flow line that runs off the left end before the homotopy turns on), and a continuation $x ⇝ z^{'}$ with $z^{'} \in Crit (f_{1})$ , $μ (z^{'}) = μ (x)$ , followed by a $\partial_{1}$ -trajectory $z^{'} \to w$ (an $f_{1}$ -flow line off the right end). The gluing theorem of 03.15.04, in its non-autonomous form, identifies these split ends with the boundary of the compact one-manifold $\overline{M^{H}} (x, w)$ . The signed count of the boundary of a compact one-manifold is zero (coherent orientations, 03.15.05); writing the two boundary types out, the first contributes $Φ^{H} \partial_{0}$ and the second contributes $\partial_{1} Φ^{H}$ , with opposite induced signs. Hence $\partial_{1} Φ^{H} - Φ^{H} \partial_{0} = 0$ .

(2) Homotopy independence. Connect $H$ to $H^{'}$ by a smooth homotopy-of-homotopies $(H^{λ})_{λ \in [0, 1]}$ with $H^{0} = H$ , $H^{1} = H^{'}$ , each constant outside a compact $s$ -interval and agreeing with the fixed endpoint pairs. Consider the parametrised moduli $W (x, y) = {(λ, u) : u \in M^{H^{λ}} (x, y)}$ . For generic such families it is a smooth manifold of dimension $μ (x) - μ (y) + 1$ . Define $K : C_{*} (f_{0}) \to C_{*+ 1} (f_{1})$ by counting, with signs, the isolated points of $W (x, y)$ in the case $μ (y) = μ (x) + 1$ , where $W (x, y)$ is zero-dimensional. The one-dimensional case $μ (y) = μ (x)$ has $W (x, y)$ a compact one-manifold whose boundary is of three types: the $λ = 0$ slice (giving $Φ^{H}$ ), the $λ = 1$ slice (giving $Φ^{H^{'}}$ ), and interior breakings (giving $\partial_{1} K$ and $K \partial_{0}$ ). The signed boundary count vanishes, which rearranges to $Φ^{H^{'}} - Φ^{H} = \partial_{1} K + K \partial_{0}$ .

(3) Composition and identity. For the constant homotopy $H_{const}$ from $(f_{0}, g_{0})$ to itself, the continuation moduli $M^{H_{const}} (x, y)$ with $μ (x) = μ (y)$ consist of constant solutions $u \equiv x$ for $x = y$ and is empty otherwise — for a Morse-Smale pair there are no nonconstant index- $0$ continuation solutions of the autonomous equation after removing the translation family. So $Φ^{H_{const}} = id$ on the nose. For concatenation, let $H_{1}$ run $(f_{0}, g_{0}) \to (f_{1}, g_{1})$ and $H_{2}$ run $(f_{1}, g_{1}) \to (f_{2}, g_{2})$ , and form the concatenation $H_{2} * H_{1}$ by placing $H_{1}$ on $s \leq 0$ and $H_{2}$ on $s \geq 0$ after a long neck. A stretching argument: insert a neck of length $T$ between the two homotopies and let $T \to \infty$ . For large $T$ the continuation moduli of $H_{2} * H_{1}$ are in oriented bijection, by gluing 03.15.04, with the fibre products $M^{H_{1}} (x, z) \times M^{H_{2}} (z, y)$ over intermediate critical points $z \in Crit (f_{1})$ with $μ (z) = μ (x) = μ (y)$ . The signed count of these is exactly the matrix product $Φ^{H_{2}} Φ^{H_{1}}$ , so the two are chain homotopic by the homotopy-of-homotopies relating $H_{2} * H_{1}$ to a small-neck representative — case (2).

Invariance. By (1), $Φ^{H}$ descends to $(Φ^{H})_{*}$ on homology. By (2), $(Φ^{H})_{*}$ is independent of $H$ , since chain-homotopic maps induce the same map on homology. By (3) and the reversed homotopy $\overset{ˉ}{H}$ , the composite $Φ^{\overset{ˉ}{H}} Φ^{H}$ is chain homotopic to $Φ^{H_{const}} = id$ , so $(Φ^{\overset{ˉ}{H}})_{*} (Φ^{H})_{*} = id$ on $H M_{*} (f_{0}, g_{0})$ , and symmetrically $(Φ^{H})_{*} (Φ^{\overset{ˉ}{H}})_{*} = id$ on $H M_{*} (f_{1}, g_{1})$ . Hence $(Φ^{H})_{*}$ is an isomorphism with inverse $(Φ^{\overset{ˉ}{H}})_{*}$ . These isomorphisms compose by (3). $□$

Bridge. This invariance theorem builds toward 03.15.08, where the common group $H M_{*} (f, g)$ — now legitimately written $H M_{*} (M)$ — is identified with the singular homology of $M$ ; the foundational reason that identification is even a well-posed question is exactly the function-independence proved here. The chain-map identity is dual to the $\partial^{2} = 0$ identity of 03.15.06: both are the statement that a one-dimensional moduli space has signed boundary zero, and the bridge is the same compact-one-manifold-with-boundary argument, run on continuation trajectories instead of on broken Morse trajectories. The homotopy-of-homotopies that produces the chain homotopy $K$ is exactly the device that reappears in 05.08.02, where continuation in the Hamiltonian governs both invariance of Floer homology and its module structure over quantum cohomology; the central insight that "two ways of counting interpolating solutions differ by the boundary of a parametrised moduli, hence by a chain homotopy" generalises verbatim from this finite-dimensional setting to every Floer theory. Putting these together, continuation identifies $H M_{*}$ of any two Morse-Smale data with one canonical group, which is exactly what licenses the notation $H M_{*} (M)$ and appears again in the functoriality of 03.15.09.

Exercises Intermediate+

Exercise 4 (medium, short-answer). The boundary of the one-dimensional continuation moduli $\overline{\mathcal{M}^{H}}(x,w)$ (with $\mu(w)=\mu(x)-1$) splits into two families: $\partial_0$-trajectory-then-continuation, and continuation-then-$\partial_1$-trajectory. Identify which of these contributes $\Phi^H \partial_0$ and which contributes $\partial_1 \Phi^H$, and explain why their signed sum is zero.

Hint

A left-end breaking happens before the homotopy turns on; a right-end breaking after it turns off. The signed boundary of a compact one-manifold vanishes.

Answer

Rubric. Full credit: the left-end breaking is an $f_{0}$ -trajectory $x \to z$ (an index- $1$ Morse line of $f_{0}$ , since the homotopy is constant $= f_{0}$ for $s ≪ 0$ ) followed by a continuation $z ⇝ w$ with $μ (z) = μ (w)$ ; summing over $z$ gives $Φ^{H} (\partial_{0} x)$ . The right-end breaking is a continuation $x ⇝ z^{'}$ with $μ (z^{'}) = μ (x)$ followed by an $f_{1}$ -trajectory $z^{'} \to w$ ; summing gives $(\partial_{1} Φ^{H}) (x)$ . By 03.15.04 gluing, these are exactly the two boundary strata of the compact one-manifold $\overline{M^{H}} (x, w)$ , and the signed count of the boundary of any compact one-manifold is $0$ (03.15.05). The two induced boundary orientations are opposite, so $Φ^{H} \partial_{0} = \partial_{1} Φ^{H}$ .

Exercise 5 (medium, numeric). On the upright sphere with one maximum and one minimum, you build a continuation from the standard height function to a slightly shifted one. How many isolated continuation trajectories run from the minimum to the new minimum, and what is the resulting matrix entry of $\Phi^H$ in that slot (with the standard orientation)?

Hint

There is exactly one index- $0$ continuation solution between the two minima.

Answer

One trajectory; entry $+ 1$ . Both minima have index $0$ , so the continuation moduli between them is zero-dimensional; for a small shift there is a single solution, oriented $+ 1$ with the standard convention. Thus $Φ^{H}$ sends the minimum generator to the new minimum generator with coefficient $+ 1$ , and likewise for the maxima — $Φ^{H}$ is the identity matrix in these bases.

Exercise 6 (hard, short-answer). Carry out the neck-stretching argument for composition: explain why, for a concatenation $H_2 \ast H_1$ with a neck of length $T \to \infty$, the index-$0$ continuation moduli are in oriented bijection with the fibre products $\bigsqcup_z \mathcal{M}^{H_1}(x,z) \times \mathcal{M}^{H_2}(z,y)$, and why this yields $\Phi^{H_2 \ast H_1} \simeq \Phi^{H_2}\Phi^{H_1}$.

Hint

As the neck grows, a solution must spend a long time near a critical point of the middle pair $(f_{1}, g_{1})$ ; that critical point is the splitting node $z$ .

Answer

Rubric. Full credit: with the neck of length $T$ , write the domain as $H_{1}$ on $s \leq - T /2$ , autonomous $(f_{1}, g_{1})$ on the neck $∣ s ∣ \leq T /2$ , and $H_{2}$ on $s \geq T /2$ . A continuation solution of total index $0$ ( $μ (x) = μ (y)$ ) must, by the compactness of 03.15.03, converge as $T \to \infty$ to a broken configuration whose middle piece is a constant solution sitting at some $z \in Crit (f_{1})$ with $μ (z) = μ (x) = μ (y)$ (any nonconstant index- $0$ neck piece is excluded after removing translation). The left and right pieces are then genuine continuation solutions in $M^{H_{1}} (x, z)$ and $M^{H_{2}} (z, y)$ . Gluing 03.15.04 inverts this: each pair $(u_{1}, u_{2})$ glues to a unique solution for large $T$ , giving the oriented bijection with $⨆_{z} M^{H_{1}} (x, z) \times M^{H_{2}} (z, y)$ . The signed count of the fibre product is the matrix product $Φ^{H_{2}} Φ^{H_{1}}$ . Finally, the long-neck $H_{2} * H_{1}$ and any fixed-neck representative are connected by a homotopy-of-homotopies, so they are chain homotopic by the independence statement, giving $Φ^{H_{2} * H_{1}} ≃ Φ^{H_{2}} Φ^{H_{1}}$ .

Exercise 7 (hard, short-answer). Show that the construction equips Morse homology with the structure of a functor from a groupoid: the objects are Morse-Smale pairs, the morphisms are homotopy classes of homotopies, and $HM_*$ sends each morphism to an isomorphism. Identify precisely which parts of the Theorem furnish identity, composition, and invertibility.

Hint

The groupoid of pairs and homotopy-classes-of-homotopies is connected and every morphism is invertible; functoriality means $Φ$ respects identity and composition up to chain homotopy.

Answer

Rubric. Full credit: define the category whose objects are Morse-Smale pairs on $M$ and whose morphisms $(f_{0}, g_{0}) \to (f_{1}, g_{1})$ are homotopy classes (rel endpoints) of admissible homotopies. Part (3) gives $Φ^{H_{const}} = id$ , so identities map to identities. Part (3) also gives $Φ^{H_{2} * H_{1}} ≃ Φ^{H_{2}} Φ^{H_{1}}$ , so composition is respected on homology. Part (2) shows $Φ^{H}$ depends only on the homotopy class, so $(Φ^{H})_{*}$ is well-defined on morphisms. The reversed homotopy $\overset{ˉ}{H}$ furnishes a two-sided inverse, so every morphism is sent to an isomorphism, making the category a groupoid and $H M_{*}$ a functor into abelian groups landing in isomorphisms. Because the groupoid is connected (any two pairs are joined by a homotopy), all the groups $H M_{*} (f, g)$ are canonically isomorphic, and the canonical-isomorphism data is the system ${(Φ^{H})_{*}}$ . Full credit identifies (3) with identity+composition, (2) with well-definedness, and $\overset{ˉ}{H}$ with invertibility.

Advanced results Master

The invariance theorem refines into a sharper functorial statement, a naturality property under smooth maps, and an explicit description of the obstruction when monotonicity is dropped.

The canonical-isomorphism system. The morphisms $(Φ^{H})_{*}$ do more than prove an abstract isomorphism: they assemble into a coherent system. For any three pairs and homotopies $H_{1}, H_{2}, H_{3}$ between them, the cocycle relation $(Φ^{H_{3}})_{*} (Φ^{H_{2}})_{*} (Φ^{H_{1}})_{*} = (Φ^{H_{3} * H_{2} * H_{1}})_{*}$ holds, and the right side depends only on the endpoints. So the inverse limit $H M_{*} (M) = lim_{(f, g)} H M_{*} (f, g)$ over the directed system of Morse-Smale pairs is a single group with canonical maps to each $H M_{*} (f, g)$ , all isomorphisms. This is the precise sense in which $H M_{*}$ is "the" homology of $M$ : not a chosen representative but the limit of the whole system, with the continuation maps as transition isomorphisms.

Naturality under embeddings and the Conley-index lineage. The continuation principle is older than Morse homology: Conley and Zehnder ^{[Conley-Zehnder 1984]} proved that the Conley index of an isolated invariant set is invariant under continuation of the flow through a family in which the set stays isolated, and Salamon ^{[Salamon 1990]} recast the Morse-theoretic case in exactly the chain-level language used here. The continuation map is the chain-level shadow of the homotopy invariance of the Conley index: the homotopy $(f_{s}, g_{s})$ is a path in the space of gradient flows, the critical points are the isolated invariant sets, and the requirement that the data be admissible throughout is the requirement that the index pair persist. Where the Conley index gives a homotopy type invariant under continuation, the Morse complex gives a chain homotopy type, and passing to homology recovers the same invariant.

Dropping monotonicity: energy and the action filtration. The chain-map identity needs no monotonicity, but quantitative refinements do. If the homotopy is monotone in the sense that $\partial_{s} f_{s} \leq 0$ pointwise, then continuation trajectories satisfy an a-priori energy bound $E (u) \leq f_{0} (x) - f_{1} (y) + \int ∥ \partial_{s} f_{s} ∥$ , and $Φ^{H}$ respects the filtration of $C_{*}$ by critical value up to a controlled shift. Without monotonicity the energy can be unbounded across the family and the filtered statement fails, even though the unfiltered chain map survives. This is the finite-dimensional germ of the role monotonicity plays in symplectic Floer theory, where it controls bubbling and underlies the construction of spectral invariants; the filtered continuation map is the engine of the spectral sequence of 03.15.10.

Independence of the metric, separately. Holding $f$ fixed and varying only the metric $g$ is the special case $f_{s} \equiv f$ , $g_{s}$ a path of metrics. The continuation map then measures how the Morse-Smale condition and the trajectory counts change as the geometry deforms; the theorem shows the homology is insensitive to $g$ as well as to $f$ . This separation matters because transversality is a condition on $(f, g)$ jointly — for a fixed $f$ one may need to perturb $g$ to achieve Morse-Smale — and invariance under the metric is what makes "the Morse homology of $f$ " well-posed even when no canonical metric is available.

Synthesis. The continuation map is the foundational device that turns a construction depending on auxiliary data into an invariant of $M$ : counting index- $0$ solutions of the non-autonomous gradient equation is exactly what produces a chain map, and the chain-map identity is dual to $\partial^{2} = 0$ of 03.15.06 in that both are the signed-boundary-is-zero statement for a compact one-manifold of trajectories. This is exactly the structure that makes $H \mapsto Φ^{H}$ a functor on the groupoid of Morse-Smale data, so that the central insight — interpolating solutions translate one complex into another, and two interpolations differ by the boundary of a parametrised moduli, hence by a chain homotopy — generalises verbatim from the finite-dimensional setting to symplectic Floer homology 05.08.02 and instanton Floer homology 03.07.23, where the same continuation argument proves invariance under the Hamiltonian or the metric. Putting these together, the canonical-isomorphism system assembles all the $H M_{*} (f, g)$ into one group $H M_{*} (M)$ , which is exactly the object 03.15.08 identifies with singular homology; the compactness of 03.15.03 that supplies the one-manifolds, the gluing of 03.15.04 that identifies their boundaries, and the coherent orientations of 03.15.05 that sign the boundary points are the three analytic inputs this functoriality consumes, and the bridge to Floer theory is the non-autonomous gradient equation, identical in form once the gradient is replaced with the gradient of the action functional.

Full proof set Master

The three identities of the Key theorem are proved in full there. The structural propositions that make the proof rigorous — finiteness of the counted sets, the index computation, and the algebraic invariance lemma — are recorded here.

Proposition (finiteness of the continuation count). For a generic homotopy $H$ and critical points $x \in Crit (f_{0})$ , $y \in Crit (f_{1})$ with $μ (x) = μ (y)$ , the moduli $M^{H} (x, y)$ is a compact zero-manifold, hence finite, and the signed count $\sum_{u} n^{H} (u)$ is a well-defined integer.

Proof. By the parametrised Sard-Smale transversality theorem applied to the non-autonomous section, for generic $H$ the operator $D_{u}^{H}$ is surjective along every solution, so $M^{H} (x, y)$ is a smooth manifold of dimension equal to $ind D_{u}^{H} = μ (x) - μ (y) = 0$ (03.15.02, spectral-flow index). Compactness follows from the non-autonomous compactness theorem (03.15.03): a sequence in $M^{H} (x, y)$ has a subsequence converging to a broken configuration $x = a_{0} ⇝ a_{1} ⇝ \dots ⇝ a_{k} = y$ ; each break costs at least one unit of index, but the total index available is $μ (x) - μ (y) = 0$ , so no break can occur and the limit is an honest solution. A compact zero-manifold is finite, and the coherent orientation (03.15.05) assigns each point a sign, so the count is a well-defined integer. $■$

Proposition (index of the continuation operator). The Fredholm index of $D_{u}^{H} = \frac{d}{d s} + A (s)$ along a continuation trajectory $u$ from $x$ to $y$ equals $μ (x) - μ (y)$ .

Proof. The operator is asymptotically constant: $A (s) \to A_{-} = Hess_{g_{0}} f_{0} (x)$ as $s \to - \infty$ and $A (s) \to A_{+} = Hess_{g_{1}} f_{1} (y)$ as $s \to + \infty$ , both invertible symmetric matrices because $x, y$ are nondegenerate. For such operators the Fredholm index equals the spectral flow of the path $A (s)$ from $A_{-}$ to $A_{+}$ , which is the signed count of eigenvalues of $A (s)$ crossing zero. Since $A_{\pm}$ are invertible, the spectral flow equals the difference of the numbers of negative eigenvalues at the two ends, $dim E^{-} (A_{-}) - dim E^{-} (A_{+}) = μ (x) - μ (y)$ , by the Robbin-Salamon spectral-flow theorem (03.15.02). The computation is independent of the interior path $A (s)$ , as it must be for the count $Φ^{H}$ to be well-defined. $■$

Proposition (chain-homotopy invariance lemma). Let $Φ, Ψ$ be chain maps between complexes of free abelian groups with $ΨΦ ≃ id$ and $ΦΨ ≃ id$ (chain homotopies). Then $Φ_{*}$ is an isomorphism, and any chain map chain-homotopic to $Φ$ induces the same map on homology.

Proof. If $Φ^{'} = Φ + \partial^{'} L + L \partial$ for a degree- $+ 1$ map $L$ , then on a cycle $c$ ( $\partial c = 0$ ), $Φ^{'} c = Φ c + \partial^{'} (L c)$ , so $[Φ^{'} c] = [Φ c]$ in homology; hence $Φ_{*}^{'} = Φ_{*}$ . Applying this to $ΨΦ ≃ id$ gives $Ψ_{*} Φ_{*} = id_{H_{*} (C)}$ , and to $ΦΨ ≃ id$ gives $Φ_{*} Ψ_{*} = id_{H_{*} (C^{'})}$ . Thus $Φ_{*}$ is invertible with inverse $Ψ_{*}$ . $■$

These three propositions supply the well-definedness, the index, and the algebra; the geometric identities (1)-(3) of the Key theorem then assemble into the invariance statement. The non-autonomous transversality and gluing inputs are the continuation-equation specialisations of the autonomous results proved in 03.15.02 and 03.15.04, and are carried out in Schwarz ^{[Schwarz Part II Ch. 5]} and Salamon ^{[Salamon 1999]}.

Connections Master

03.15.06 (the Morse complex and $\partial^{2} = 0$ ) is the construction this unit makes invariant. The chain map identity $\partial_{1} Φ^{H} = Φ^{H} \partial_{0}$ is proved by the same compact-one-manifold-with-boundary argument that proves $\partial^{2} = 0$ there, run on continuation trajectories rather than on a single function's broken trajectories. The boundary operators $\partial_{0}, \partial_{1}$ whose intertwining is asserted are exactly the ones built in that unit.

03.15.08 (the Morse Homology Theorem) consumes this invariance directly: once $H M_{*} (f, g)$ is known to be independent of $(f, g)$ , it may be written $H M_{*} (M)$ and compared with singular homology by choosing a convenient self-indexing Morse function. Without continuation invariance the statement " $H M_{*} (M) ≅ H_{*} (M)$ " would not even be well-posed, since the left side would depend on a choice.

03.15.10 (Poincaré duality and the filtered spectral sequence) refines the unfiltered continuation map of this unit into a filtered one. The monotone homotopies discussed in the Advanced results respect the filtration of $C_{*} (f)$ by critical value, and the resulting filtered continuation maps are the maps of spectral sequences that make the filtered invariant well-defined.

05.08.02 (Floer homology) is the infinite-dimensional heir: continuation maps for homotopies of Hamiltonians prove that symplectic Floer homology is independent of the Hamiltonian and almost-complex structure, by the identical chain-map-and-chain-homotopy argument, with the non-autonomous gradient equation replaced by the $s$ -dependent Floer equation. The functoriality proved here is the literal template for the Hamiltonian-invariance theorem there.

03.07.23 (instanton Floer homology) uses the same continuation principle to prove metric-independence of the instanton homology of a homology three-sphere; the homotopy-of-homotopies producing the chain homotopy $K$ is the finite-dimensional model of the cobordism-of-cobordisms argument used in the gauge-theoretic setting.

Historical & philosophical context Master

The continuation principle predates Morse homology as a chain-level construction. Charles Conley's index theory, developed through the 1970s and crystallised in the work of Conley and Eduard Zehnder, Morse-type index theory for flows and periodic solutions for Hamiltonian equations (Communications on Pure and Applied Mathematics 37, 1984, 207–253), established that the homotopy type carried by an isolated invariant set persists under continuation of the flow, provided the set remains isolated throughout. This homotopy invariance under deformation is the topological ancestor of the chain map $Φ^{H}$ : the Morse complex is the chain-level refinement of the Conley index, and the continuation map is the chain-level refinement of the index's continuation isomorphism ^{[Conley-Zehnder 1984]}.

The translation of this principle into the explicit chain-homotopy language used for the Morse-Smale-Witten complex is due to Dietmar Salamon, Morse theory, the Conley index and Floer homology (Bulletin of the London Mathematical Society 22, 1990, 113–140), written in the immediate wake of Floer's construction of his complex; Andreas Floer had already used continuation maps in the symplectic and instanton settings, in Witten's complex and infinite-dimensional Morse theory (Journal of Differential Geometry 30, 1989, 207–221) ^{[Floer 1989]}. Matthias Schwarz's Morse Homology (Progress in Mathematics 111, Birkhäuser, 1993), from his 1992 ETH Zürich dissertation under Zehnder and Salamon, carried out the finite-dimensional continuation construction in full analytic detail in Part II Chapter 5, as the template every Floer theory reproduces. The non-autonomous gradient equation Schwarz analyses is the exact finite-dimensional shadow of Floer's $s$ -dependent equation, which is why the proofs transfer line for line to the infinite-dimensional setting that motivated them.

Bibliography Master

@book{Schwarz1993,
  author    = {Schwarz, Matthias},
  title     = {Morse Homology},
  series    = {Progress in Mathematics},
  volume    = {111},
  publisher = {Birkh\"auser Verlag, Basel},
  year      = {1993}
}

@article{Salamon1990,
  author  = {Salamon, Dietmar},
  title   = {Morse theory, the {C}onley index and {F}loer homology},
  journal = {Bulletin of the London Mathematical Society},
  volume  = {22},
  number  = {2},
  pages   = {113--140},
  year    = {1990}
}

@article{ConleyZehnder1984,
  author  = {Conley, Charles and Zehnder, Eduard},
  title   = {Morse-type index theory for flows and periodic solutions for {H}amiltonian equations},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {37},
  number  = {2},
  pages   = {207--253},
  year    = {1984}
}

@article{Floer1989,
  author  = {Floer, Andreas},
  title   = {Witten's complex and infinite-dimensional {M}orse theory},
  journal = {Journal of Differential Geometry},
  volume  = {30},
  number  = {1},
  pages   = {207--221},
  year    = {1989}
}

@incollection{Salamon1999,
  author    = {Salamon, Dietmar},
  title     = {Lectures on {F}loer homology},
  booktitle = {Symplectic Geometry and Topology (Park City, UT, 1997)},
  series    = {IAS/Park City Mathematics Series},
  volume    = {7},
  pages     = {143--229},
  publisher = {American Mathematical Society, Providence, RI},
  year      = {1999}
}

@book{AudinDamian2014,
  author    = {Audin, Mich\`ele and Damian, Mihai},
  title     = {Morse Theory and Floer Homology},
  series    = {Universitext},
  publisher = {Springer-Verlag, London},
  year      = {2014}
}

Prerequisites

03.15.03
03.15.04
03.15.05

Tier anchors

beginner: Audin-Damian *Morse Theory and Floer Homology* §3.3 (continuation and invariance); Banyaga-Hurtubise *Lectures on Morse Homology* Ch. 11
intermediate: Schwarz *Morse Homology* Part II Ch. 5; Audin-Damian §3.3; Salamon *Lectures on Floer homology* §3
master: Schwarz *Morse Homology* Part II Ch. 5; Salamon 1990; Conley-Zehnder 1984

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Gradient flows, deformation and the invariance of the Morse picture under change of function (background)
Schwarz — Morse Homology (Progress in Mathematics 111, Birkhäuser 1993) · Part II, Ch. 5 (continuation homomorphisms and invariance of the Morse homology)
Salamon — Morse theory, the Conley index and Floer homology · Bulletin of the London Mathematical Society 22 (1990), pp. 113-140 (continuation and the invariance principle)
Conley-Zehnder — Morse-type index theory for flows and periodic solutions for Hamiltonian equations · Communications on Pure and Applied Mathematics 37 (1984), pp. 207-253 (the continuation principle for the Conley index)
Floer — Witten's complex and infinite-dimensional Morse theory · Journal of Differential Geometry 30 (1989), pp. 207-221 (continuation maps in the infinite-dimensional setting)
Salamon — Lectures on Floer homology · IAS/Park City Mathematics Series 7 (1999), §3 (chain homotopies and invariance)

Estimated time

beginner: 16m
intermediate: 42m
master: 78m