03.15.11 · modern-geometry / morse-homology

From finite-dimensional Morse homology to Floer homology

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Anchor (Master): Salamon *Lectures on Floer Homology* §1-3; Floer 1988 (J. Diff. Geom. 28); Salamon-Zehnder 1992 (Comm. Pure Appl. Math. 45); Hofer-Salamon 1995 (Floer homology and Novikov rings)

Intuition Beginner

For the whole of this chapter you built a homology out of a height function on a curved shape. One slot for each resting point of the height, sorted by how many directions the height falls away, and a boundary rule that counts the flow streams running from one resting point down to the next. The streams were the gradient lines of the height, and the homology they produced turned out to be the ordinary homology of the shape. The whole machine ran on three facts: there are finitely many resting points, the streams between adjacent ones are finite in number, and the moduli of streams have tidy ends.

The bridge this unit describes takes that exact machine and points it at a space that is no longer a finite-dimensional shape. Instead of a curved surface, the new playground is a space of loops, or a space of fields. Instead of a height function with finitely many resting points, there is a quantity — an action — whose resting points are the special loops or special fields you actually care about in physics and geometry. The hope is that the same bookkeeping machine still works.

It does not work for free. A space of loops is infinitely many dimensions tall, so a resting point can have infinitely many downhill directions and infinitely many uphill ones. The Morse index, the count of downhill directions, becomes the difference of two infinities and has to be redefined. Yet the streams between resting points stay finite-dimensional and well-behaved, and that is the miracle that lets the machine survive.

Visual Beginner

Alt text: A two-column figure. The left column shows the finite-dimensional Morse setup: a torus with a height function, its resting points, short flow arrows, and a box stack reading "one slot per resting point." The right column shows the Floer setup: a tube standing for the space of loops, special loops marked as resting points of an action, long connecting cylinders, and a box stack reading "one slot per special loop." A central bracket reads "same machine: resting points + flow streams," with arrows pairing height with action, downhill direction with spectral shift, and stream with cylinder. The Floer construction is the Morse machine run on an infinite-dimensional space.

Worked example Beginner

Take the simplest infinite-dimensional playground: the loops on a flat doughnut, the torus you have met all chapter. A loop is a way of walking around the torus and returning to where you started. The space of all such loops is huge — to name one loop you must give a position for every instant of the walk — so it has infinitely many dimensions.

Now put a quantity on this space. Roughly, score each loop by how much it winds and how much energy the walk spends. The loops that cannot lower their score by any small wiggle are the resting points. On the flat torus these are the loops that run straight around at constant speed; the constant loops, sitting still at one point, are among them. These special loops play the role the resting points of a height function played before.

What about the streams? A stream is now a way of continuously deforming one special loop into another while always sliding downhill in score. Drawn out in time, such a deformation is not a curve but a surface: a tube, or cylinder, swept out as the loop moves. So the flow streams of the finite picture become flowing cylinders here.

Counting these cylinders between special loops of adjacent grade, with signs, gives a boundary rule, and its homology is the Floer homology of the torus. The lesson is not the final number but the shape of the analogy: resting loops in place of resting points, cylinders in place of streams, the same counting machine on top.

Check your understanding Beginner

Question 2 (easy, multiple-choice). Why does the count of downhill directions (the Morse index) need a new definition in the Floer setting?

Because there are no resting points at all in infinite dimensions.
Because a resting point can have infinitely many downhill and infinitely many uphill directions, so the plain count is a difference of two infinities.
Because the action is always negative.
Because loops cannot be deformed continuously.

Hint

What goes wrong with "count the downhill directions" when the space is infinitely many dimensions tall?

Answer

B. In an infinite-dimensional space a resting point typically has infinitely many directions of each kind, so the raw count of downhill directions is not a finite number and must be replaced by a relative grading. Feedback if correct: exactly — the index becomes a comparison between resting points rather than an absolute count. Feedback if wrong: the issue is that both the downhill and uphill counts are infinite, so a finite Morse index has to be redefined as a difference.

Question 3 (medium, short-answer). Name the three structural facts about the finite-dimensional Morse machine that the Floer construction needs to keep, and state in one phrase which one is hardest to recover in infinite dimensions.

Hint

Finitely many resting points, finitely many streams between adjacent grades, tidy ends for the stream moduli. Which of these can break when surfaces are involved?

Answer

Sample answer. The three facts are: finitely many resting points of the relevant grade, finitely many flow streams between resting points one grade apart, and well-behaved (compact, tidy) ends of the stream moduli. The hardest to recover is the tidy-ends fact, because flowing cylinders can degenerate in a new way special to surfaces — a small bubble of area can pinch off — which has no finite-dimensional analogue. Feedback if correct: right, the new failure mode is bubbling, absent from the finite theory. Feedback if wrong: the finite ingredients are the resting-point count, the stream count, and the tidy ends; the extra difficulty is bubbling of the flowing cylinders.

Formal definition Intermediate+

Fix a closed symplectic manifold $(M, ω)$ and a time-dependent Hamiltonian $H : S^{1} \times M \to R$ . Let $L M = C^{\infty} (S^{1}, M)$ be the free loop space, and on a suitable cover $L M$ define the symplectic action functional $A_{H}$ whose first variation is $$ d\mathcal{A}H(\gamma),\xi ;=; \int_0^1 \omega\bigl(\dot\gamma(t) - X{H_t}(\gamma(t)),, \xi(t)\bigr),dt , $$ so the critical points of $A_{H}$ are exactly the $1$ -periodic orbits of the Hamiltonian flow, $\overset{γ}{˙} = X_{H_{t}} (γ)$ . This is the infinite-dimensional Morse function: $L M$ replaces the closed manifold of 03.15.01, and the orbits replace the critical points. In the gauge-theoretic incarnation the loop space is replaced by the orbit space $B (Y) = A (Y) / G (Y)$ of connections modulo gauge on a $3$ -manifold $Y$ , and $A_{H}$ is replaced by the Chern-Simons functional, whose critical points are flat connections (see 03.07.23).

Choosing an $ω$ -compatible almost complex structure $J$ gives an $L^{2}$ metric on $L M$ , and the formal negative $L^{2}$ -gradient flow of $A_{H}$ is the Floer equation for maps $u : R \times S^{1} \to M$ , $$ \partial_s u + J(u)\bigl(\partial_t u - X_{H_t}(u)\bigr) ;=; 0 , $$ an elliptic Cauchy-Riemann-type partial differential equation ^{[Floer 1988]}. A finite gradient trajectory of 03.15.01 becomes a pseudoholomorphic cylinder $u$ connecting two orbits $x = lim_{s \to - \infty} u$ and $y = lim_{s \to + \infty} u$ ; in the gauge theory the same role is played by an anti-self-dual instanton on the cylinder $R \times Y$ .

The grading is the point where the dictionary forces a change. The Morse index of 03.15.01 counts negative eigenvalues of the Hessian, but at a Hamiltonian orbit the Hessian of $A_{H}$ has infinitely many eigenvalues of each sign. The replacement is the Conley-Zehnder index $μ_{C Z} (x)$ of 05.08.04, defined as the spectral flow of the family of self-adjoint operators obtained by linearising along the orbit — the net signed count of eigenvalues crossing zero. Only the difference $μ_{C Z} (x) - μ_{C Z} (y)$ is intrinsic, and it equals the Fredholm index of the linearised Floer operator $D_{u}$ , the analogue of the spectral-flow index of 03.15.02. The relative grading replaces the absolute Morse index.

When $(M, ω)$ is not exact, the action $A_{H}$ is multivalued on $L M$ (it changes by periods of $ω$ along loops in $L M$ ), and there can be infinitely many trajectories of the same relative index. The coefficient ring is then enlarged to a Novikov ring $Λ$ — a ring of formal series in a variable tracking the symplectic area, with the finiteness condition that only finitely many terms occur below any energy bound ^{[Hofer-Salamon 1995]}. This restores the finiteness of the boundary count that 03.15.06 obtained for free.

Counterexamples to common slips

The Floer equation is not an honest gradient flow: $A_{H}$ has no well-defined finite negative-gradient vector field, the flow does not exist for all time on $L M$ , and the unstable manifolds of orbits are infinite-dimensional. Only the trajectory spaces $M (x, y)$ — the moduli of finite-energy solutions — are finite-dimensional, and that is what the construction uses; the phrase " $A_{H}$ is a Morse function" is a structural analogy, not a literal statement.
The Conley-Zehnder index is not a Morse index in disguise: there is in general no canonical absolute integer, only a relative one, and even that depends on a choice of trivialisation along a capping disk. The Maslov class of $M$ measures the dependence, and for monotone $M$ the grading lives in $Z /2 N$ , not $Z$ .
The compactification of $M (x, y)$ adds two kinds of boundary, not one: broken cylinders, exactly as in 03.15.03, and bubbled configurations where a holomorphic sphere splits off. The sphere bubbles have no finite-dimensional counterpart; transversality or a monotonicity hypothesis is needed to push them into high codimension so that $\partial^{2} = 0$ survives.
Novikov coefficients are not a cosmetic relabelling: over a field they can change the rank of Floer homology, and the spectral sequence from the energy filtration (the action analogue of the critical-value filtration of 03.15.10) need not collapse. The non-exact theory is genuinely richer than its finite-dimensional template.

Key theorem with proof Intermediate+

The content of the bridge is that, with the redefinitions above, the chain-level machine of 03.15.06 runs verbatim and produces a homology that is an invariant of $(M, ω)$ .

Theorem (Floer's construction, dictionary form). Let $(M, ω)$ be a closed monotone symplectic manifold and $H$ a Hamiltonian with nondegenerate $1$ -periodic orbits, $J$ generic. Define $CF_(H) = \bigoplus_x \Lambda\langle x\rangle $g r a d e d m o d$ 2N$ by the Conley-Zehnder index, with* $$ \partial x ;=; \sum_{\mu_{CZ}(x)-\mu_{CZ}(y)=1} \Bigl(\textstyle\sum_{u\in\widehat{\mathcal{M}}(x,y)} n(u), T^{\mathcal{A}(u)}\Bigr) y . $$ Then $\partial^{2} = 0$ , and the resulting Floer homology $HF_(M,\omega) $i s in d e p e n d e n t o f$ H $an d$ J $, w i t h$ HF_*(M,\omega) \cong H_{+n}(M;\Lambda)$.

Proof (dictionary translation). Each ingredient is the named finite-dimensional unit with one analytic upgrade. Trajectory moduli. The space $M (x, y)$ of finite-energy solutions of the Floer equation is the zero set of a Fredholm section of a Banach bundle over a Banach manifold of cylinders in weighted Sobolev spaces, and its expected dimension is the index of $D_{u}$ , computed as the spectral flow $μ_{C Z} (x) - μ_{C Z} (y)$ — the verbatim analogue of 03.15.02, with the Conley-Zehnder index of 05.08.04 in place of the Morse index. Transversality. For generic $J$ (or generic Hamiltonian perturbation) the linearised operator is surjective along every solution, so $M (x, y)$ is a smooth manifold of the expected dimension; this is the Sard-Smale Banach-genericity argument of 03.15.02, applied to the universal moduli space. Compactness. A sequence of solutions with bounded energy has a subsequence converging to a broken cylinder, exactly as in 03.15.03, unless energy concentrates at a point, in which case a holomorphic sphere bubbles off; monotonicity makes the bubble locus codimension at least two, so the index-one and index-two moduli are unaffected. Gluing and orientations. The gluing of 03.15.04 and the coherent orientations of 03.15.05 carry over without change, producing a manifold-with-boundary structure on the compactified index-two moduli and characteristic signs $n (u)$ .

The identity $\partial^{2} = 0$ . With Novikov coefficients enforcing the finiteness that 03.15.06 obtained for free, the coefficient of $y$ in $\partial^{2} x$ is the signed count of the boundary of a compact one-manifold and vanishes. Invariance. The continuation maps of 03.15.07 — built from the Floer equation with an $s$ -dependent Hamiltonian — show $H F_{*}$ is independent of $(H, J)$ . The final isomorphism with $H_{*+ n} (M; Λ)$ is obtained by taking $H$ a small autonomous Morse function, whose orbits are its critical points and whose low-energy cylinders are honest Morse trajectories; the Floer complex degenerates to the Morse complex of 03.15.06, and the Morse Homology Theorem 03.15.08 supplies the identification. $□$

Bridge. This construction builds toward the two Floer chapters already in the corpus and is the reason they read as variations on a single theme; the same dictionary appears again in 05.08.02 (symplectic Floer homology) and in 03.07.23 (instanton Floer homology), where the loop space is replaced by a gauge-orbit space and the action by the Chern-Simons functional. The foundational reason the finite-dimensional theory is the right template is that every analytic input of Floer's machine — Fredholm index, transversality, compactness, gluing, orientations — was already isolated as a self-standing unit in this chapter, so the infinite-dimensional theory is exactly those units with the Morse index replaced by spectral flow. This is exactly the substitution recorded in the grading: the central insight is that the Conley-Zehnder index of 05.08.04 is the spectral-flow index of 03.15.02 read in infinite dimensions, and putting these together, the only genuinely new phenomenon — bubbling — is what separates the Floer theory from a literal copy of 03.15.08. The bridge is that the Morse Homology Theorem itself does not survive verbatim, because no finite cell structure exists on $L M$ ; what survives is the continuation invariance of 03.15.07, and that is precisely the statement the Floer units prove.

Exercises Intermediate+

Exercise 1 (easy, short-answer). State, for each of the five finite-dimensional units [03.15.02]-[03.15.06], its infinite-dimensional Floer counterpart in one phrase.

Hint

Trajectory spaces / Fredholm; compactness; gluing; orientations; the complex with $\partial^{2} = 0$ .

Answer

Sample answer. 03.15.02 (trajectory spaces, Fredholm, Sard-Smale) becomes the Banach manifold of finite-energy cylinders with the linearised Floer operator, index = spectral flow, transversality for generic $J$ . 03.15.03 (broken-trajectory compactness) becomes Gromov compactness: broken cylinders plus sphere bubbling. 03.15.04 (gluing) carries over verbatim as the gluing of cylinders. 03.15.05 (coherent orientations) carries over as the orientation of determinant lines of the Floer operator. 03.15.06 ( $\partial^{2} = 0$ ) carries over, now over a Novikov ring to enforce finiteness.

Exercise 3 (medium, short-answer). Explain why the grading of $CF_*$ is only relative, and what extra datum is needed to lift it to an absolute $\mathbb{Z}$- or $\mathbb{Z}/2N$-grading.

Hint

The Hessian of $A_{H}$ has infinitely many eigenvalues of each sign; what does spectral flow measure, and relative to what trivialisation?

Answer

Sample answer. The Hessian of $A_{H}$ at an orbit is an unbounded self-adjoint operator with infinitely many eigenvalues of each sign, so it has no finite Morse index. Spectral flow assigns an integer only to a path between two such operators, so only the difference $μ_{C Z} (x) - μ_{C Z} (y)$ is canonical. To fix an absolute index one trivialises the symplectic vector bundle $γ^{*} T M$ along a capping disk of each orbit; the Conley-Zehnder index of the resulting path of symplectic matrices is then an integer, well-defined modulo the Maslov numbers of $π_{2} (M)$ . For monotone $M$ this descends to a $Z /2 N$ -grading with $2 N$ the minimal Chern number.

Exercise 4 (medium, short-answer). Why is a Novikov ring needed in the non-exact case, and what finiteness does it restore?

Hint

In the non-exact case there can be infinitely many cylinders of the same relative index; how do their areas distribute?

Answer

Sample answer. When $ω$ is not exact, $A_{H}$ is multivalued and two orbits can be joined by infinitely many cylinders of the same Fredholm index but differing symplectic areas. The naive integer count $\sum_{u} n (u)$ is then an infinite sum and ill-defined. The Novikov ring $Λ$ records each cylinder with a formal weight $T^{A (u)}$ and imposes the condition that, below any energy bound, only finitely many weights occur. This makes the matrix coefficient $\sum_{u} n (u) T^{A (u)}$ a well-defined Novikov series, restoring the finiteness that the compact manifold gave automatically in 03.15.06.

Exercise 5 (hard, short-answer). Sketch how taking $H$ a small autonomous Morse function degenerates the Floer complex of $(M,\omega)$ to the Morse complex of [03.15.06], and identify the index shift.

Hint

For small $H$ the only $1$ -periodic orbits are the critical points (constant orbits); compute the Conley-Zehnder index of a constant orbit in terms of the Morse index.

Answer

Rubric. Full credit: for $H = ϵ f$ with $f$ Morse and $ϵ$ small, every nonconstant $1$ -periodic orbit has period bounded below, so the only orbits are the constant ones at the critical points of $f$ . For a constant orbit at a critical point $p$ the linearised Hamiltonian flow is $exp (t ϵ Hess f (p))$ , a small rotation, and the Conley-Zehnder index is $μ_{C Z} (p) = μ_{Morse} (p) - n$ (with $dim M = 2 n$ ), the Morse index shifted by $- n$ . Low-energy Floer cylinders between such orbits are, by an estimate of Floer-Hofer-Salamon, in bijection with the Morse gradient lines of $f$ , so the Floer differential reduces to the Morse differential of 03.15.06. Hence $H F_{*} (M, ω) ≅ H M_{*+ n} (f) ≅ H_{*+ n} (M; Λ)$ by the Morse Homology Theorem 03.15.08, explaining both the isomorphism and the degree shift by $n$ .

Exercise 6 (hard, short-answer). Identify which single phenomenon obstructs a naive transcription of the $\partial^2 = 0$ proof of [03.15.06] to the Floer setting, and state two standard hypotheses that control it.

Hint

What new boundary stratum can appear in the compactified index-two moduli that has no finite-dimensional analogue?

Answer

Rubric. Full credit: the obstruction is bubbling. The boundary of the compactified index-two moduli $\overline{M} (x, z)$ contains, besides the once-broken cylinders that cancel as in 03.15.06, configurations where a holomorphic sphere of positive area splits off, carrying away an index that breaks the pairing. Two standard controls: (i) monotonicity, $[ω] = λ c_{1} (T M)$ with $λ > 0$ , which forces the sphere-bubble locus into codimension $\geq 2$ so the index-one and index-two strata see no bubbles; (ii) for the general case, an abstract perturbation scheme (virtual fundamental cycles / Kuranishi structures or polyfolds) that achieves transversality even where bubbling occurs. With either hypothesis the boundary count is again that of a compact one-manifold and $\partial^{2} = 0$ .

Advanced results Master

The dictionary is not a single isomorphism but a family of parallel constructions, each obtained by choosing the infinite-dimensional space and the action; the finite-dimensional chapter is the common rehearsal for all of them.

The two incarnations. In the symplectic incarnation the playground is the loop space $L M$ , the action is $A_{H}$ , its critical points are Hamiltonian orbits, and the flow lines are pseudoholomorphic cylinders; this is the Floer homology of 05.08.02, with grading by the Conley-Zehnder index of 05.08.04. In the instanton incarnation the playground is the gauge-orbit space $B (Y) = A (Y) / G (Y)$ over a homology $3$ -sphere $Y$ , the action is the Chern-Simons functional, its critical points are flat connections, and the flow lines are anti-self-dual instantons on the cylinder $R \times Y$ ; this is the instanton Floer homology of 03.07.23, graded mod $8$ by the spectral flow of 03.07.19. The two share every structural step and differ only in the analysis: Gromov compactness with sphere bubbling on one side, Uhlenbeck compactness with energy bubbling and the reducible-connection subtlety on the other.

Spectral flow as the universal index. The single conceptual replacement that powers both incarnations is the passage from the Morse index — a count of negative Hessian eigenvalues, finite because the manifold is finite-dimensional — to spectral flow, the net number of eigenvalues of the linearised operator that cross zero along a path of self-adjoint operators. The Atiyah-Patodi-Singer theory identifies this spectral flow with the Fredholm index of the associated cylinder operator, so the expected dimension of $M (x, y)$ is computed by an index theorem rather than by counting directions. The Conley-Zehnder index 05.08.04 and the mod- $8$ instanton grading 03.07.19 are two specialisations of one spectral-flow recipe.

Novikov rings and the energy filtration. In the non-exact case the action filtration of $C F_{*}$ by the value $A_{H} (x)$ is the infinite-dimensional analogue of the critical-value filtration of 03.15.10; its associated spectral sequence has $E_{1}$ page the Novikov-graded count of orbits and converges to $H F_{*}$ , with the continuation maps as maps of spectral sequences exactly as in the finite case. The Novikov ring is what makes the boundary count finite term-by-term, and the resulting Floer homology is a module over it whose rank can detect the non-exactness invisible to the underlying manifold's ordinary homology.

Where the analogy stops. The Morse Homology Theorem 03.15.08 had two proofs; only one survives. The cellular route needs a finite CW structure from descending cells, and the descending manifolds in $L M$ are infinite-dimensional, so that route has no analogue. The axiomatic route survives as the continuation invariance of 03.15.07, which is exactly the statement the Floer chapters prove: $H F_{*}$ is an invariant of the geometric data, not of the auxiliary $(H, J)$ . The further identification $H F_{*} (M, ω) ≅ H_{*+ n} (M; Λ)$ is the Arnold-conjecture payoff, available only because the small-Hamiltonian limit reconnects to the finite-dimensional Morse complex.

Synthesis. The foundational reason the finite-dimensional Morse chapter is built before either Floer chapter is that Floer homology is, line for line, the Morse machine of 03.15.02-03.15.06 with the manifold replaced by an infinite-dimensional configuration space and the Morse index replaced by spectral flow; this is exactly the substitution that the grading records, and the central insight is that nothing in transversality, compactness, gluing, or orientations changes its logical role in the passage. Putting these together, the only genuinely new analytic phenomenon is bubbling, which is why the compactness unit 03.15.03 is the one whose Floer analogue acquires an extra boundary stratum, and the only structural casualty is the cellular half of the Morse Homology Theorem 03.15.08, which is dual to the surviving axiomatic half exported as continuation invariance 03.15.07. The bridge is that the Conley-Zehnder index of 05.08.04 and the instanton grading of 03.07.19 are one spectral-flow construction generalising the Morse index, so the two Floer chapters 05.08.02 and 03.07.23 are two readings of a single dictionary whose original is this chapter; this same pattern recurs whenever an action functional replaces a height function, and it is the reason Schwarz wrote the finite-dimensional theory with full analytic detail rather than as a corollary of classical Morse theory.

Full proof set Master

The Key theorem section gives the dictionary translation in outline. The propositions below pin down the two load-bearing claims: that the relative index is the spectral flow, and that the small-Hamiltonian limit recovers the Morse complex.

Proposition (the Floer index is a spectral flow). *Let $x, y$ be nondegenerate $1$ -periodic orbits and $u$ a finite-energy solution of the Floer equation asymptotic to $x$ at $- \infty$ and $y$ at $+ \infty$ . After trivialising $u^{*} T M$ , the linearised operator $D_{u} = \partial_{s} + J_{0} \partial_{t} + S (s, t)$ on weighted Sobolev spaces is Fredholm, and its index equals the spectral flow of the path of self-adjoint operators $A_{s} = - J_{0} \partial_{t} - S (s, \cdot)$ on $L^{2} (S^{1}, R^{2 n})$ , which equals $μ_{C Z} (x) - μ_{C Z} (y)$ .*

Proof. The asymptotic operators $A_{\pm \infty}$ are the Hessians of $A_{H}$ at $x$ and $y$ ; nondegeneracy means $0$ is not in their spectrum, so $D_{u}$ has invertible ends and is Fredholm on the weighted spaces by the standard estimate for Cauchy-Riemann operators with invertible asymptotics ^{[Salamon-Zehnder 1992]}. The Fredholm index of such an operator equals the spectral flow of the connecting path ${A_{s}}$ — the net signed count of eigenvalues crossing zero — by the index identity for self-adjoint families (the cylinder case of the Atiyah-Patodi-Singer theorem). Finally, $μ_{C Z}$ is defined so that the spectral flow from $A_{- \infty}$ to $A_{+ \infty}$ through the reference operator is $μ_{C Z} (x) - μ_{C Z} (y)$ ; this is the Conley-Zehnder index of 05.08.04. The Morse-theoretic content is that this number plays the exact role the difference of Morse indices played in 03.15.02, so the moduli $M (x, y)$ has expected dimension $μ_{C Z} (x) - μ_{C Z} (y)$ . $■$

Proposition (small Hamiltonians recover the Morse complex). Let $f$ be a Morse function on $M$ and $H_{ϵ} = ϵ f$ (time-independent), with $g = ω (\cdot, J \cdot)$ the metric. For $ϵ > 0$ small enough the $1$ -periodic orbits of $X_{H_{ϵ}}$ are precisely the constant orbits at the critical points of $f$ , each nondegenerate, with $μ_{C Z} (p) = μ_{Morse} (p) - n$ ; and there is a bijection between rigid Floer cylinders and Morse gradient lines, so $CF_(H_\epsilon) \cong C_{+n}(f)$ as complexes.

Proof. A nonconstant $1$ -periodic orbit of $X_{ϵ f}$ has its period bounded below by a constant over $ϵ$ times the $C^{2}$ -norm of $f$ (the orbit must traverse a definite amount of the flow), so for small $ϵ$ no nonconstant orbit closes up in time one; only the zeros of $X_{ϵ f}$ , i.e. the critical points of $f$ , remain, and each is nondegenerate because $Hess f$ is. At a constant orbit the linearised flow is $exp (t ϵ J Hess f (p))$ , a small loop of symplectic matrices winding through the negative eigenspace of $Hess f$ ; its Conley-Zehnder index is the Morse index shifted by the constant $- n$ coming from the normalisation of $μ_{C Z}$ for the constant path. For the differential, a theorem of Floer-Hofer-Salamon shows that for small $ϵ$ the finite-energy Floer cylinders of index one are, after rescaling, uniformly close to and in bijection with the negative-gradient trajectories of $f$ , with matching signs; the bubble locus is empty in this energy range. Hence the Floer differential equals the Morse differential of 03.15.06 under $p \mapsto p$ , and $C F_{*} (H_{ϵ}) ≅ C_{*+ n} (f)$ . Passing to homology and invoking 03.15.08 gives $H F_{*} (M, ω) ≅ H_{*+ n} (M; Λ)$ . $■$

Proposition (continuation invariance is the surviving half of the Morse Homology Theorem). The continuation maps built from the $s$ -dependent Floer equation induce isomorphisms $HF_(H_0,J_0)\cong HF_*(H_1,J_1) $s a t i s f y in g t h es am e f u n c t or ia l i t y (i d e n t i t y an d co m p os i t i o n u pt oc hainh o m o t o p y) a s t h e M or seco n t in u a t i o nma p so f [03.15.07] . C o n se q u e n tl y$ HF_*(M,\omega)$ is well-defined, and this is the only part of 03.15.08 that transfers to infinite dimensions.*

Proof. The construction is the verbatim image of 03.15.07: interpolate $(H_{0}, J_{0})$ to $(H_{1}, J_{1})$ by an $s$ -dependent family, count index-zero solutions of the resulting non-autonomous Floer equation to define a chain map, and use the index-one solutions of a homotopy of homotopies to build the chain homotopies proving functoriality. Compactness for these maps is again Gromov compactness with the bubble locus pushed to high codimension by monotonicity. The cellular route of 03.15.08 cannot be imitated because $L M$ carries no finite CW structure from descending sets, those being infinite-dimensional; thus the absolute identification with a cell complex is unavailable, and invariance under $(H, J)$ — the homotopy-invariance axiom of the axiomatic route — is what remains. This is exactly the statement 05.08.02 and 03.07.23 establish. $■$

These three propositions are the analytic spine of the bridge: the first supplies the grading, the second supplies the comparison to the finite-dimensional theory, and the third isolates precisely which theorem of this chapter does and does not survive.

Connections Master

03.15.08 (the Morse Homology Theorem) is the template this unit deforms. Its two proofs are the fork in the road: the cellular route dies in infinite dimensions for want of a finite CW structure, while the axiomatic route survives as continuation invariance. The bridge keeps the half of 03.15.08 that is a universal property and discards the half that is a cell count, and the small-Hamiltonian limit is what reconnects the surviving invariant to ordinary homology with Novikov coefficients.

03.15.07 (continuation maps and invariance of $H M_{*}$ ) is the finite-dimensional rehearsal of the only invariance that Floer homology can claim. In the Morse setting continuation shows independence of the pair; in the Floer setting the identical construction — index-zero solutions of an $s$ -dependent equation, chain homotopies from a homotopy of homotopies — shows independence of $(H, J)$ , and is the precise content that 05.08.02 and 03.07.23 prove. The continuation theory is the part of the chapter that scales to infinite dimensions without amendment.

05.08.02 (symplectic Floer homology) is one of the two destinations of the dictionary, the loop-space incarnation with action $A_{H}$ and pseudoholomorphic cylinders. This unit supplies the finite-dimensional original that 05.08.02 is the analogue of; conversely 05.08.02 carries out the analysis — Gromov compactness, monotonicity, the Arnold-conjecture isomorphism — that this bridge can only name.

05.08.04 (the Conley-Zehnder index) is the grading that replaces the Morse index. The bridge identifies it as the spectral-flow specialisation that makes the relative index of 03.15.02 meaningful when the Hessian has infinite index, and the small-Hamiltonian Proposition computes it as the Morse index shifted by $- n$ , which is the source of the degree shift in $H F_{*} ≅ H_{*+ n}$ .

03.07.19 (spectral flow and the Floer grading mod 8) and 03.07.23 (instanton Floer homology) are the gauge-theoretic incarnation of the same dictionary, with $L M$ replaced by the orbit space $B (Y)$ and $A_{H}$ by the Chern-Simons functional. They show the bridge is not specific to symplectic geometry: the same spectral-flow grading and the same compactness-gluing-orientation spine run through anti-self-dual instantons, with energy bubbling and reducibles as the analytic novelties.

03.15.10 (Poincaré duality via flow reversal; the filtered Morse spectral sequence) is the finite-dimensional source of the action filtration. Its critical-value filtration becomes the energy filtration of $C F_{*}$ over a Novikov ring, with the same spectral sequence and the same continuation maps as maps of spectral sequences, so the filtered machinery of this chapter transfers alongside the unfiltered complex.

Historical & philosophical context Master

The idea that an action functional should be treated as a Morse function on an infinite-dimensional space is older than Floer — it is implicit in Marston Morse's own work on the calculus of variations in the large and explicit in Raoul Bott's Morse-theoretic proof of periodicity — but the obstacle was always that the index is infinite and the gradient flow does not exist. The decisive move was Andreas Floer's, in The unregularized gradient flow of the symplectic action (Communications on Pure and Applied Mathematics 41, 1988, 775-813) ^{[Floer 1988]} and Morse theory for Lagrangian intersections (Journal of Differential Geometry 28, 1988, 513-547) ^{[Floer 1988b]}: do not regularise the flow on the loop space at all, but study only the finite-energy solutions of the elliptic equation that the formal flow would satisfy, and grade by a relative index. Edward Witten's Supersymmetry and Morse theory (1982) had already recast the finite-dimensional differential as an instanton count and gestured at the infinite-dimensional version, and Floer's construction made that gesture into a theorem.

The grading by spectral flow rather than Morse index was clarified by Dietmar Salamon and Eduard Zehnder in Morse theory for periodic solutions of Hamiltonian systems and the Maslov index (Communications on Pure and Applied Mathematics 45, 1992, 1303-1360) ^{[Salamon-Zehnder 1992]}, which identified the relative Floer index with the Conley-Zehnder index and proved the Arnold conjecture for monotone manifolds. The coefficient question — what to do when the action is multivalued — was settled by Helmut Hofer and Salamon in Floer homology and Novikov rings (in The Floer Memorial Volume, 1995, 483-524) ^{[Hofer-Salamon 1995]}, importing the Novikov ring from the topology of closed one-forms. Matthias Schwarz's Morse Homology (1993), written in this same circle at ETH Zürich under Zehnder and Salamon, was designed precisely as the finite-dimensional rehearsal: by carrying out the trajectory-moduli analysis with full rigour on a closed manifold, it isolated exactly the steps that Floer's theory would reuse and exposed, by their absence, the one step — the cellular comparison — that infinite dimensions would forbid. The philosophical lesson the dictionary teaches is that a homology theory need not come from a space at all: it can come from a functional, provided its critical points are isolated and its flow lines are compact enough to count.

Bibliography Master

@article{Floer1988action,
  author  = {Floer, Andreas},
  title   = {The unregularized gradient flow of the symplectic action},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {41},
  number  = {6},
  pages   = {775--813},
  year    = {1988}
}

@article{Floer1988lagrangian,
  author  = {Floer, Andreas},
  title   = {Morse theory for {L}agrangian intersections},
  journal = {Journal of Differential Geometry},
  volume  = {28},
  number  = {3},
  pages   = {513--547},
  year    = {1988}
}

@article{SalamonZehnder1992,
  author  = {Salamon, Dietmar and Zehnder, Eduard},
  title   = {Morse theory for periodic solutions of {H}amiltonian systems and the {M}aslov index},
  journal = {Communications on Pure and Applied Mathematics},
  volume  = {45},
  number  = {10},
  pages   = {1303--1360},
  year    = {1992}
}

@incollection{HoferSalamon1995,
  author    = {Hofer, Helmut and Salamon, Dietmar A.},
  title     = {Floer homology and {N}ovikov rings},
  booktitle = {The Floer Memorial Volume},
  series    = {Progress in Mathematics},
  volume    = {133},
  publisher = {Birkh\"auser, Basel},
  pages     = {483--524},
  year      = {1995}
}

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

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

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

@article{Witten1982,
  author  = {Witten, Edward},
  title   = {Supersymmetry and {M}orse theory},
  journal = {Journal of Differential Geometry},
  volume  = {17},
  number  = {4},
  pages   = {661--692},
  year    = {1982}
}

Prerequisites

03.15.08

Tier anchors

beginner: Audin-Damian *Morse Theory and Floer Homology* §6.1 (the action functional as an infinite-dimensional Morse function, informal); Schwarz *Morse Homology* Introduction
intermediate: Salamon *Lectures on Floer Homology* (IAS/Park City) §1-2; Audin-Damian Part II Ch. 8-11; Schwarz *Morse Homology* Introduction and Part III
master: Salamon *Lectures on Floer Homology* §1-3; Floer 1988 (J. Diff. Geom. 28); Salamon-Zehnder 1992 (Comm. Pure Appl. Math. 45); Hofer-Salamon 1995 (Floer homology and Novikov rings)

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · The variational (action-functional) viewpoint on Morse theory and the path-space picture that underlies the loop-space analogy (background)
Schwarz — Morse Homology (Progress in Mathematics 111, Birkhäuser 1993) · Introduction and Part III (the finite-dimensional theory as the deliberate model for Floer's construction; the action-functional analogy and the isolated trajectory-moduli analysis)
Floer — The unregularized gradient flow of the symplectic action · Communications on Pure and Applied Mathematics 41 (1988), pp. 775-813 (the symplectic action functional, its L^2 gradient flow as Floer's equation, and the resulting homology)
Floer — Morse theory for Lagrangian intersections · Journal of Differential Geometry 28 (1988), pp. 513-547 (the Maslov index grading and the relative index of intersection points)
Salamon-Zehnder — Morse theory for periodic solutions of Hamiltonian systems and the Maslov index · Communications on Pure and Applied Mathematics 45 (1992), pp. 1303-1360 (the Conley-Zehnder index as spectral flow; the Fredholm theory of the linearised Cauchy-Riemann operator)
Hofer-Salamon — Floer homology and Novikov rings · in The Floer Memorial Volume (Progress in Mathematics 133, Birkhäuser 1995), pp. 483-524 (the Novikov ring as the coefficient system enforcing finiteness in the non-exact case)
Salamon — Lectures on Floer homology · in Symplectic Geometry and Topology (IAS/Park City Math. Series 7, AMS 1999), §1-3 (the systematic Morse-to-Floer dictionary)

Estimated time

beginner: 18m
intermediate: 46m
master: 84m