03.15.06 · modern-geometry / morse-homology

The Morse complex and $\partial^{2} = 0$

shipped3 tiersLean: none

Anchor (Master): Schwarz *Morse Homology* Part II Ch. 1, 4; Witten 1982; Thom 1949

Intuition Beginner

You have spent several units learning to count the downhill streams between resting points of a height function, and to attach a plus or minus sign to each one. Now you assemble those signed counts into a single bookkeeping machine. The machine is built from one slot for each resting point, sorted into shelves by how many downhill directions that resting point has. A bottom has zero downhill directions; a saddle has one; a top has two. The shelves are the rows of the machine.

The machine has one operation, a boundary rule. Feed it a resting point, and it returns the signed total of streams running down to the resting points one shelf below. So the boundary of a top is the signed count of streams from the top to the saddles; the boundary of a saddle is the signed count of streams from that saddle to the bottoms. The whole structure of the height function is now encoded in this one rule.

The surprise, and the heart of the unit, is that applying the boundary rule twice always gives nothing at all. Start at a top, take its boundary to reach the saddles, take the boundary again to reach the bottoms, and everything cancels to zero. This is not an accident of any one example. It is forced by a clean geometric fact about how streams between far-apart shelves can break apart and rejoin, which you met in the gluing and compactness units. Because applying the rule twice gives zero, the machine has a homology, and that homology turns out to recover the holes of the underlying shape.

Visual Beginner

Alt text: A three-row column diagram for the height function on an upright torus: a bottom box b in the lowest row, two saddle boxes s1 and s2 in the middle row, a top box t in the upper row. Signed downward arrows connect the rows, with the two streams into each saddle and the two streams into the bottom carrying opposite signs that cancel. A side panel shows the composite top-to-bottom routes cancelling, so the boundary rule applied twice yields zero. The picture conveys the chain group as boxes sorted by index and the boundary-squared-is-zero identity.

Worked example Beginner

Take the upright doughnut once more: a top $t$ , two saddles $s_{1}$ and $s_{2}$ partway down, and a bottom $b$ . Sort the four resting points by their number of downhill directions. The bottom $b$ has none, so it sits on shelf $0$ . The two saddles each have one, so they sit on shelf $1$ . The top $t$ has two, so it sits on shelf $2$ .

Now compute the boundary rule. From the top $t$ to each saddle there are two downhill streams, and the signs you assigned in the orientations unit make the two streams to $s_{1}$ cancel and the two streams to $s_{2}$ cancel. So the boundary of $t$ is $0$ . From each saddle down to the bottom $b$ there are again two streams, and again they come in opposite-sign pairs, so the boundary of $s_{1}$ is $0$ and the boundary of $s_{2}$ is $0$ .

Apply the rule twice to the top. First step: the boundary of $t$ is $0$ . Second step: the boundary of $0$ is $0$ . So the rule applied twice gives $0$ , as promised.

What this tells us: counting the slots that survive cancellation, shelf $0$ keeps one combination, shelf $1$ keeps two, shelf $2$ keeps one. The numbers $1, 2, 1$ are exactly the number of independent holes of the doughnut in each dimension: one piece, two loops, one cavity. The machine has recovered the shape from nothing but the height function and its signed stream counts.

Check your understanding Beginner

Formal definition Intermediate+

Fix a closed Riemannian manifold $(M^{n}, g)$ and a Morse-Smale pair $(f, g)$ , with critical set $Crit (f)$ , Morse index $μ (x)$ , unparametrized trajectory spaces $M (x, y) = M (x, y) / R$ , and a fixed coherent orientation furnishing the characteristic sign $n (u) \in {\pm 1}$ of each isolated flow line, exactly as in 03.15.01, 03.15.02, and 03.15.05.

The Morse chain group is the free abelian group on the critical points, graded by index: $$ C_k(f) = \bigoplus_{x \in \mathrm{Crit}(f),\ \mu(x) = k} \mathbb{Z}\langle x\rangle, \qquad C_*(f) = \bigoplus_{k} C_k(f). $$ Because $M$ is compact and $f$ is Morse, $Crit (f)$ is finite, so each $C_{k} (f)$ is a finitely generated free abelian group. For $μ (x) - μ (y) = 1$ the moduli $M (x, y)$ is a compact $0$ -manifold, hence a finite set of isolated flow lines, and the signed count is the integer $$ n(x,y) = \sum_{u \in \widehat{\mathcal{M}}(x,y)} n(u) \in \mathbb{Z}. $$

The Morse boundary operator $\partial = \partial_{k} : C_{k} (f) \to C_{k - 1} (f)$ is defined on generators by $$ \partial\langle x\rangle = \sum_{\substack{y \in \mathrm{Crit}(f)\ \mu(y) = \mu(x) - 1}} n(x,y),\langle y\rangle, $$ and extended $Z$ -linearly. The pair $(C_{*} (f), \partial)$ is the Morse-Smale-Witten complex of $(f, g)$ . When the identity $\partial \circ \partial = 0$ holds (the Key theorem below), the Morse homology is $$ HM_k(f,g) = \frac{\ker(\partial_k : C_k \to C_{k-1})}{\operatorname{im}(\partial_{k+1} : C_{k+1} \to C_k)}. $$ The sign convention is fixed once and for all by the chosen coherent orientation of 03.15.05; a different coherent orientation conjugates $\partial$ by a diagonal sign-change $x \mapsto ε (x) x$ and leaves $H M_{*}$ unchanged.

Counterexamples to common slips

The boundary operator counts only flow lines between adjacent indices, $μ (x) - μ (y) = 1$ . A pair with $μ (x) - μ (y) = 0$ has $M (x, y)$ empty (no nonconstant flow line lowers nothing), and a pair with $μ (x) - μ (y) \geq 2$ has positive-dimensional moduli that contribute no isolated points to count; these do not enter $\partial$ .
The signed count $n (x, y)$ is an integer, not a residue mod $2$ . Over $Z /2$ one counts flow lines without signs and $\partial^{2} = 0$ already holds; the content of the coherent-orientation theory is that the signed integer count also squares to zero, giving the $Z$ -coefficient complex.
$H M_{*} (f, g)$ is the homology of the complex, not the chain group itself. The chain group remembers every critical point; the homology remembers only the holes. The doughnut has the chain ranks $1, 2, 1$ , which happen to equal its Betti numbers because its height function is perfect — but a height function with extra cancelling critical points has larger chain ranks and the same homology.
A coherent orientation is required to define $n (x, y)$ over $Z$ . Dropping it and taking absolute counts gives a $Z /2$ complex; declaring a sign per flow line without the gluing-compatibility of 03.15.05 gives a $\partial$ that need not square to zero.

Key theorem with proof Intermediate+

The identity $\partial^{2} = 0$ is the structural payoff of Parts I and II of Schwarz ^{[Schwarz Part II Ch. 4]}. It is what turns the boundary operator into the differential of a genuine chain complex.

Theorem ( $\partial^{2} = 0$ ). Let $(f, g)$ be a Morse-Smale pair on a closed Riemannian manifold $(M^{n}, g)$ with a fixed coherent orientation. Then $\partial_{k - 1} \circ \partial_{k} = 0$ for every $k$ ; equivalently, for all critical points $x, z$ with $μ (x) - μ (z) = 2$ , $$ \sum_{\substack{y \in \mathrm{Crit}(f)\ \mu(y) = \mu(x) - 1}} n(x,y),n(y,z) = 0. $$

Proof. Expanding the definition on a generator $⟨ x ⟩$ with $μ (x) = k$ , $$ \partial^2\langle x\rangle = \partial\Big(\sum_{\mu(y)=k-1} n(x,y)\langle y\rangle\Big) = \sum_{\mu(z)=k-2}\Big(\sum_{\mu(y)=k-1} n(x,y),n(y,z)\Big)\langle z\rangle, $$ so the claim is that the coefficient of each $⟨ z ⟩$ with $μ (x) - μ (z) = 2$ vanishes. Fix such a pair $(x, z)$ and consider the unparametrized moduli $M (x, z)$ , which by transversality 03.15.02 is a smooth $1$ -manifold of dimension $μ (x) - μ (z) - 1 = 1$ .

By the compactness theorem 03.15.03, the closure $\overline{M} (x, z)$ is compact, and the only limit points added are broken trajectories. A break at an intermediate critical point $y$ requires $μ (x) > μ (y) > μ (z)$ , so $μ (y) = μ (x) - 1$ , and a once-broken trajectory is a pair $(u, v)$ with $u \in M (x, y)$ , $v \in M (y, z)$ , each in a $0$ -dimensional moduli, hence a discrete pair. Triple breaks are excluded because they would need two intermediate indices strictly between $k$ and $k - 2$ , of which there is only one available. By the gluing theorem 03.15.04, each once-broken trajectory $(u, v)$ is the endpoint of exactly one half-open arc of honest trajectories, and conversely every end of every arc limits to one such broken trajectory. Therefore $\overline{M} (x, z)$ is a compact $1$ -manifold with boundary, and its boundary is precisely the finite set of once-broken trajectories $$ \partial,\overline{\widehat{\mathcal{M}}}(x,z) = \bigsqcup_{\mu(y)=k-1}\ \widehat{\mathcal{M}}(x,y) \times \widehat{\mathcal{M}}(y,z). $$

A compact $1$ -manifold with boundary is a finite disjoint union of circles and closed arcs. Circles have no boundary; each closed arc has exactly two boundary points. With the coherent orientation of 03.15.05 inducing an orientation on the $1$ -manifold and hence on its boundary, the two endpoints of any arc receive opposite induced signs — this is exactly the statement, proved in 03.15.05, that coherence forces the two ends of a glued arc to carry opposite characteristic signs, where the sign at an end $(u, v)$ equals the product $n (u) n (v)$ . Summing the induced boundary signs over all arcs therefore gives zero: $$ \sum_{(u,v) \in \partial,\overline{\widehat{\mathcal{M}}}(x,z)} n(u),n(v) = 0. $$ Grouping the boundary points by their intermediate critical point $y$ and using $n (x, y) = \sum_{u} n (u)$ , $n (y, z) = \sum_{v} n (v)$ , the left side is $\sum_{μ (y) = k - 1} n (x, y) n (y, z)$ , which is the coefficient of $⟨ z ⟩$ in $\partial^{2} ⟨ x ⟩$ . Hence that coefficient vanishes for every $z$ , so $\partial^{2} ⟨ x ⟩ = 0$ . $□$

Bridge. This theorem builds toward 03.15.07 and 03.15.08, where the homology $H M_{*} (f, g)$ of the complex is shown invariant under the Morse-Smale pair and identified with singular homology. The foundational reason $\partial^{2} = 0$ holds is the boundary of a boundary: the algebraic identity is the signed count of the boundary of a compact $1$ -manifold, which is always zero, and this is exactly the finite-dimensional shadow of the relation $\partial^{2} = 0$ in singular homology, where the same vanishing reflects that the boundary of a simplex has no boundary. Putting these together, the compactness of 03.15.03, the gluing of 03.15.04, and the coherent orientations of 03.15.05 are not three unrelated technical results: they are precisely the three facts needed to make the boundary of $\overline{M} (x, z)$ a finite signed set whose total is zero. The central insight is that the differential of the Morse complex is a boundary count, so its square is a count of a boundary-of-a-boundary, and the identity is forced rather than checked example by example. This pattern appears again in 03.07.23 and 05.08.02, where the same arc-counting argument over the compactified index- $2$ moduli gives $\partial^{2} = 0$ for the instanton and symplectic Floer differentials; the bridge is that the compactified one-dimensional moduli is a $1$ -manifold with boundary in every one of these theories, and only the moduli's analytic origin differs.

Exercises Intermediate+

Exercise 5 (medium, short-answer). Explain why no flow line is counted in $\partial$ between two critical points of equal index, and why this is consistent with $\dim\widehat{\mathcal{M}}(x,y) = \mu(x)-\mu(y)-1$.

Hint

What is the dimension of $M (x, y)$ when $μ (x) = μ (y)$ ?

Answer

Rubric. Full credit: when $μ (x) = μ (y)$ the formula gives $dim M (x, y) = - 1$ , which means the moduli is empty — there is no nonconstant negative-gradient trajectory between distinct critical points of the same index under Morse-Smale transversality, since such a trajectory would lie in $W^{u} (x) \cap W^{s} (y)$ , a transverse intersection of dimension $μ (x) - μ (y) = 0$ in $M$ would be needed but the unparametrized count subtracts one. The boundary operator only counts the $0$ -dimensional unparametrized moduli, which occur exactly at index difference one. Same-index pairs contribute nothing, matching the empty negative-dimensional moduli.

Exercise 6 (hard, short-answer). State precisely which property of the compactified moduli $\overline{\widehat{\mathcal{M}}}(x,z)$ ($\mu(x)-\mu(z)=2$) makes the coefficient of $\langle z\rangle$ in $\partial^2\langle x\rangle$ vanish, and explain how compactness, gluing, and coherent orientations each contribute one piece of that property.

Hint

The vanishing is the signed boundary count of a compact $1$ -manifold with boundary.

Answer

Rubric. Full credit: the coefficient equals the signed count of the boundary of $\overline{M} (x, z)$ , and the signed boundary of a compact $1$ -manifold with boundary is zero (each closed arc has two oppositely-oriented endpoints; circles contribute none). Compactness 03.15.03 supplies that $\overline{M} (x, z)$ is compact and that its added points are once-broken trajectories $(u, v)$ with $μ (y) = μ (x) - 1$ (no triple breaks, since only one intermediate index is available). Gluing 03.15.04 supplies that each broken trajectory is exactly one boundary point — one end of one arc — so the boundary set is in bijection with the index- $1$ broken trajectories and the space is a genuine manifold-with-boundary. Coherent orientations 03.15.05 supply that the two ends of each arc carry opposite signs, with the sign at $(u, v)$ equal to $n (u) n (v)$ , so the boundary sum is $\sum_{(u, v)} n (u) n (v) = \sum_{y} n (x, y) n (y, z)$ and it vanishes. Each is necessary: without compactness the boundary set is not finite; without gluing it is not a manifold boundary; without coherent signs the count is unsigned and only $Z /2$ -zero.

Exercise 7 (hard, short-answer). Compute the Morse complex of the height function on the torus $T^2$ standing upright (one maximum $t$ at index $2$, two saddles $s_1, s_2$ at index $1$, one minimum $b$ at index $0$) and verify that $HM_*(T^2) = (\mathbb{Z}, \mathbb{Z}^2, \mathbb{Z})$.

Hint

For this standard height function every signed count $n (\cdot, \cdot)$ between adjacent indices is zero.

Answer

Rubric. Full credit: chain groups $C_{2} = Z ⟨ t ⟩$ , $C_{1} = Z ⟨ s_{1} ⟩ \oplus Z ⟨ s_{2} ⟩$ , $C_{0} = Z ⟨ b ⟩$ . From $t$ to each saddle there are two flow lines that, with the coherent orientation, carry opposite signs, so $n (t, s_{1}) = n (t, s_{2}) = 0$ and $\partial_{2} = 0$ . From each saddle to $b$ there are likewise two flow lines of opposite sign, so $n (s_{1}, b) = n (s_{2}, b) = 0$ and $\partial_{1} = 0$ . With both differentials zero, $H M_{2} = C_{2} = Z$ , $H M_{1} = C_{1} = Z^{2}$ , $H M_{0} = C_{0} = Z$ , matching $H_{*} (T^{2}; Z)$ . The height function on $T^{2}$ is perfect: chain ranks $1, 2, 1$ equal the Betti numbers.

Lean formalization Intermediate+

Mathlib carries free abelian groups, $Z$ -graded chain complexes (HomologicalComplex, ChainComplex), and the homology of a complex, so the algebraic target — a graded complex with $\partial^{2} = 0$ and its homology — is available once the differential is built. What is missing is the geometric construction of the differential and the proof that it squares to zero, which rests on the coherent-orientation and compactified-moduli layers absent from Mathlib, so this unit ships at lean_status: none. The pseudo-Lean below indicates the intended shape; it does not compile because MorseSmale, signedCount, and unparamTrajectorySpace are not defined in Mathlib.

-- Illustrative only; not wired into the Lean build (lean_status: none).

-- The graded Morse chain group: free abelian on index-k critical points.
noncomputable def morseChain (g) (f) [MorseSmale g f] (k : ℕ) : Type _ :=
  FreeAbelianGroup {x : M // IsCrit f x ∧ index f x = k}

-- The signed count n(x,y) = Σ_{u} charSign u, for μ(x)-μ(y)=1.
noncomputable def signedCount (o : CoherentOrientation g f) (x y : M)
    (h : index f x - index f y = 1) : ℤ :=
  Finset.sum (unparamTrajectorySpace g f x y).toFinset (charSign o x y)

-- The boundary operator ∂x = Σ_{μ(y)=μ(x)-1} n(x,y) y.
noncomputable def morseBoundary (o : CoherentOrientation g f) (k : ℕ) :
    morseChain g f k →+ morseChain g f (k-1) := sorry

-- The central identity ∂² = 0, from the signed boundary of the compact
-- 1-manifold Mbar(x,z) with μ(x)-μ(z)=2.
theorem morseBoundary_sq (o : CoherentOrientation g f) (k : ℕ) :
    (morseBoundary o (k-1)).comp (morseBoundary o k) = 0 := by sorry

-- Morse homology as the homology of the resulting complex.
noncomputable def morseHomology (o : CoherentOrientation g f) (k : ℕ) : Type _ :=
  sorry  -- ker (morseBoundary o k) / im (morseBoundary o (k+1))

Once morseBoundary is supplied and morseBoundary_sq is proved, the homology and the weak Morse inequality rank (morseChain) ≥ rank (morseHomology) follow from Mathlib's existing HomologicalComplex and homology API; the named gap is the geometric differential and the signed-boundary-count theorem, not the algebra of complexes.

Advanced results Master

Beyond the bare identity, the Morse complex carries a precise algebraic structure: a $Z$ -grading by index, an immediate Morse inequality, an Euler-characteristic identity, and an invariance under the orientation data that makes the homology canonical.

The Euler characteristic is the Poincaré-Hopf count. The graded Euler characteristic of the complex is $$ \chi(C_*(f), \partial) = \sum_k (-1)^k \operatorname{rank} C_k(f) = \sum_{x \in \mathrm{Crit}(f)} (-1)^{\mu(x)}. $$ Because $\partial$ has degree $- 1$ , the alternating sum of chain ranks equals the alternating sum of homology ranks, $\sum_{k} (- 1)^{k} c_{k} = \sum_{k} (- 1)^{k} b_{k} = χ (M)$ . The right-hand side is the signed count of critical points, which is the Poincaré-Hopf theorem for the gradient field; the Morse complex recovers it as the Euler characteristic of a complex whose homology is that of $M$ . The identity holds before any orientation is chosen, since it depends only on the index counts, not on the signs $n (x, y)$ .

The strong Morse inequalities. Writing the Poincaré polynomial of the homology $P_{t} = \sum_{k} b_{k} t^{k}$ and the Morse counting polynomial $M_{t} = \sum_{k} c_{k} t^{k}$ , the existence of the complex $(C_{*}, \partial)$ with $H_{*} (C_{*}) = H M_{*}$ gives $M_{t} - P_{t} = (1 + t) Q_{t}$ for a polynomial $Q_{t}$ with non-negative integer coefficients, where $Q_{t}$ records the ranks of the boundary maps. This is the polynomial form of the strong Morse inequalities, and it follows from a standard fact in homological algebra: for any finite complex of finitely generated free abelian groups, the counting polynomial and the Poincaré polynomial differ by $(1 + t)$ times a non-negative polynomial. The Morse complex realizes the inequalities of 03.02.31 as the algebra of an honest differential, where Milnor's handle argument produced them by sublevel-set induction.

Witten's deformation predicts the differential. Witten's deformed de Rham operator $d_{t} = e^{- t f} d e^{t f}$ , acting on differential forms, has the same cohomology as $d$ for every $t$ , since conjugation by an invertible operator preserves cohomology ^{[Witten 1982]}. As $t \to \infty$ , the spectrum of the associated Laplacian $Δ_{t}$ concentrates: the small eigenvalues are separated from the large ones by a gap of order $t$ , and the eigenforms with small eigenvalue localize, one for each critical point, near that point — a copy of the local harmonic oscillator ground state in the unstable directions. The finite-dimensional subcomplex spanned by these low-lying eigenforms is isomorphic to the Morse complex, and the off-diagonal matrix elements of $d_{t}$ between adjacent critical points are, to leading order as $t \to \infty$ , the tunnelling amplitudes along the gradient trajectories connecting them — exactly the signed counts $n (x, y)$ . This analytic derivation, made rigorous by Helffer-Sjöstrand and, in the moduli-counting form, by Floer ^{[Floer 1989]}, is the physical origin of the complex of this unit.

Orientation-independence of the homology. A change of coherent orientation recorded by $ε : Crit (f) \to {\pm 1}$ conjugates the differential, $\partial^{'} = T \partial T^{- 1}$ with $T ⟨ x ⟩ = ε (x) ⟨ x ⟩$ , so the two complexes are isomorphic and $H M_{*} (f, g)$ is independent of the orientation choice up to canonical isomorphism. Combined with the invariance under the Morse-Smale pair established by continuation in 03.15.07, the homology depends only on $M$ , which is the content of the Morse Homology Theorem 03.15.08.

Synthesis. The boundary operator is the foundational object: it is a signed boundary count, and the entire chapter exists to make that count well-defined and to prove its square vanishes. Putting these together, the identity $\partial^{2} = 0$ is the signed cardinality of the boundary of a compact $1$ -manifold, which is exactly the finite-dimensional incarnation of the relation $\partial^{2} = 0$ in singular homology — the bridge is that both express "the boundary of a boundary is empty," one for compactified trajectory moduli and one for chains of simplices, and the Morse Homology Theorem 03.15.08 is the statement that these two boundary-of-a-boundary relations compute the same homology. The Euler-characteristic identity generalises the Poincaré-Hopf theorem; the strong inequalities are dual to the handle filtration of 03.02.31; and the Witten deformation predicts the differential analytically, which is the central insight that the same complex arises from spectral concentration as from trajectory counting. This pattern appears again verbatim in 03.07.23 and 05.08.02: replacing the gradient flow by the anti-self-duality or Cauchy-Riemann flow leaves the arc-counting proof of $\partial^{2} = 0$ unchanged, and Schwarz's finite-dimensional complex is, by design, the rehearsal that makes those infinite-dimensional differentials intelligible.

Full proof set Master

The identity $\partial^{2} = 0$ is proved in full in the Key theorem section. The supporting structural propositions are recorded here.

Proposition (well-definedness and finiteness of $\partial$ ). For a Morse-Smale pair $(f, g)$ with a fixed coherent orientation, the boundary operator $\partial : C_{k} (f) \to C_{k - 1} (f)$ is a well-defined homomorphism of finitely generated free abelian groups.

Proof. Since $M$ is compact and $f$ is Morse, $Crit (f)$ is finite (the critical points are isolated by the Morse lemma 03.02.30 and a compact manifold admits finitely many isolated points), so each $C_{k} (f)$ is a finite-rank free abelian group. For $μ (x) - μ (y) = 1$ the moduli $M (x, y)$ has dimension $μ (x) - μ (y) - 1 = 0$ by transversality 03.15.02, and it is compact: the compactification 03.15.03 adds only broken trajectories, which require an intermediate critical point of index strictly between $μ (x)$ and $μ (y)$ , of which there are none when the indices differ by one. A compact $0$ -manifold is finite, so the signed count $n (x, y) = \sum_{u} n (u)$ is a finite sum of $\pm 1$ 's, a well-defined integer. Extending linearly over the finite generating set defines $\partial$ on all of $C_{k} (f)$ . $■$

Proposition (Euler characteristic). For a Morse-Smale pair on a closed $M$ , $\sum_{k} (- 1)^{k} rank C_{k} (f) = \sum_{k} (- 1)^{k} rank H M_{k} (f, g)$ , and both equal $\sum_{x} (- 1)^{μ (x)}$ .

Proof. For any bounded complex of finitely generated free abelian groups $(C_{*}, \partial)$ , the rank-nullity theorem applied to each $\partial_{k} : C_{k} \to C_{k - 1}$ gives $rank C_{k} = rank ker \partial_{k} + rank im \partial_{k}$ . Writing $z_{k} = rank ker \partial_{k}$ , $b_{k} = rank im \partial_{k + 1}$ , the homology rank is $rank H M_{k} = z_{k} - b_{k}$ (a subquotient of free abelian groups has rank equal to the difference of ranks). Then $$ \sum_k (-1)^k \operatorname{rank} C_k = \sum_k(-1)^k(z_k + b_{k-1}) = \sum_k (-1)^k(z_k - b_k) = \sum_k (-1)^k \operatorname{rank} HM_k, $$ where the middle step reindexes $b_{k - 1}$ and the alternating signs convert the $+ b$ into $- b$ . The left side is by definition $\sum_{x} (- 1)^{μ (x)}$ , since $rank C_{k}$ is the number of index- $k$ critical points. $■$

Proposition (perfect functions and the lower bound). If a Morse-Smale pair has $\partial = 0$ , then $rank C_{k} (f) = rank H M_{k} (f, g) = b_{k} (M)$ for all $k$ ; conversely, any Morse function whose critical points have no two indices differing by one has $\partial = 0$ and is perfect.

Proof. If $\partial = 0$ then $ker \partial_{k} = C_{k}$ and $im \partial_{k + 1} = 0$ , so $H M_{k} = C_{k}$ and the chain ranks equal the homology ranks; once $H M_{*} ≅ H_{*} (M)$ 03.15.08, these are the Betti numbers $b_{k} (M)$ . For the converse, the boundary operator only counts flow lines between indices differing by one; if no two critical indices differ by one, every $n (x, y)$ in the defining sum is a count over an empty index-pair, so $\partial = 0$ and the function is perfect by the first part. The standard examples are the round height functions on $CP^{n}$ (critical points in even indices $0, 2, \dots, 2 n$ ) and on $S^{n}$ (indices $0$ and $n$ only, for $n \geq 2$ ). $■$

The three propositions are proved here in full; the strong Morse inequalities in polynomial form follow from the standard $(1 + t)$ -divisibility of $M_{t} - P_{t}$ for finite free complexes, and the identification $H M_{*} ≅ H_{*} (M)$ is the subject of 03.15.08 and Schwarz Part II ^{[Schwarz Part II Ch. 1]}.

Connections Master

03.15.03 (compactness, broken trajectories) supplies the first half of the boundary identification driving $\partial^{2} = 0$ . Its theorem that $\overline{M} (x, z)$ is compact with added points exactly the broken trajectories — and that for index difference two these are once-broken with a single intermediate index — is what makes the boundary of the index- $2$ moduli a finite set to be signed and summed. Without compactness the boundary count would be infinite and meaningless.

03.15.04 (gluing of trajectories) supplies the second half: each once-broken trajectory is exactly one boundary point of $\overline{M} (x, z)$ , the endpoint of one half-open arc. This upgrades the compact space to a compact $1$ -manifold with boundary, whose boundary is in bijection with the index- $1$ broken trajectories. Gluing is what turns "compact" into "manifold-with-boundary," the property whose signed boundary count vanishes.

03.15.05 (coherent orientations and characteristic signs) supplies the signs. Its theorem that coherence forces the two ends of each glued arc to carry opposite characteristic signs, with the sign at a broken end $(u, v)$ equal to $n (u) n (v)$ , is exactly what makes the signed boundary count $\sum_{(u, v)} n (u) n (v) = \sum_{y} n (x, y) n (y, z)$ vanish over $Z$ rather than only over $Z /2$ . The whole purpose of coherence was to make this integral differential square to zero.

03.02.31 (handle attachment, CW homotopy type, Morse inequalities) is the classical counterpart computing the same homology by sublevel-set induction. There a $λ$ -cell is attached at each index- $λ$ critical point and the Morse inequalities arise from the cellular chain complex of the resulting CW structure; here the same inequalities arise from an intrinsic differential counting flow lines, and the Morse Homology Theorem 03.15.08 identifies the two answers.

03.07.23 (instanton Floer homology) is the infinite-dimensional gauge-theoretic analogue: the Floer differential counts anti-self-dual instantons on a cylinder connecting flat connections, and $\partial^{2} = 0$ is proved by the identical argument — the compactified index- $2$ instanton moduli is a $1$ -manifold with boundary whose ends are once-broken instanton trajectories that cancel in signed pairs. Schwarz designed the finite-dimensional complex of this unit as the rehearsal for exactly this construction.

05.08.02 (Floer homology) inherits the same arc-counting proof verbatim in the symplectic setting, with pseudoholomorphic strips in place of gradient flow lines; the boundary operator and its squaring to zero are the loop-space transcription of this unit's finite-dimensional complex.

Historical & philosophical context Master

The idea that the critical points of a function, graded by index and connected by gradient trajectories, should assemble into a chain complex computing the homology of the manifold was first articulated as a physical mechanism by Edward Witten in Supersymmetry and Morse theory (Journal of Differential Geometry 17, 1982, 661–692), where the deformed differential $d_{t} = e^{- t f} d e^{t f}$ is shown to have its small eigenforms localize at critical points as $t \to \infty$ , and the off-diagonal matrix elements between adjacent critical points are computed as tunnelling amplitudes along the connecting gradient trajectories. Witten's paper predicted the Morse-Smale-Witten complex and its differential but left the analytic justification of the instanton count as a programme. The cell-decomposition picture underlying the chain group — the partition of $M$ into descending (unstable) cells, one of dimension $μ (x)$ per critical point — goes back to René Thom, Sur une partition en cellules associée à une fonction sur une variété (C.R. Acad. Sci. Paris 228, 1949, 973–975) ^{[Thom 1949]}, and to Smale's work on the Morse-Smale transversality condition in the 1960s.

The rigorous moduli-counting realization of Witten's complex — defining the differential by signed counts of gradient trajectories and proving $\partial^{2} = 0$ from the compactified one-dimensional moduli — was carried out by Andreas Floer in the course of his Floer-homology programme, notably Witten's complex and infinite-dimensional Morse theory (Journal of Differential Geometry 30, 1989, 207–221) ^{[Floer 1989]}, in the infinite-dimensional setting of the symplectic action functional. The first self-contained finite-dimensional account, with the trajectory moduli as honest finite-dimensional manifolds and the full chain of transversality, compactness, gluing, and coherent orientations assembled into a clean proof that $\partial^{2} = 0$ and that $H M_{*} ≅ H_{*} (M; Z)$ , is Matthias Schwarz's Morse Homology (Progress in Mathematics 111, Birkhäuser, 1993) ^{[Schwarz Part II Ch. 1]}, from his 1992 ETH Zürich dissertation under Eduard Zehnder and Dietmar Salamon, where the finite-dimensional complex is presented as the deliberate finite-dimensional rehearsal for Floer's infinite-dimensional original.

Bibliography Master

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

@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}
}

@article{Thom1949,
  author  = {Thom, Ren\'e},
  title   = {Sur une partition en cellules associ\'ee \`a une fonction sur une vari\'et\'e},
  journal = {Comptes Rendus de l'Acad\'emie des Sciences, Paris},
  volume  = {228},
  pages   = {973--975},
  year    = {1949}
}

@article{Floer1989witten,
  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}
}

@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}
}

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

@book{BanyagaHurtubise2004,
  author    = {Banyaga, Augustin and Hurtubise, David},
  title     = {Lectures on Morse Homology},
  series    = {Kluwer Texts in the Mathematical Sciences},
  volume    = {29},
  publisher = {Kluwer Academic Publishers, Dordrecht},
  year      = {2004}
}

Prerequisites

03.15.03
03.15.04
03.15.05

Tier anchors

beginner: Banyaga-Hurtubise *Lectures on Morse Homology* Ch. 7 (the Morse-Smale-Witten complex, informal); Audin-Damian *Morse Theory and Floer Homology* §4.1 (the complex and $\partial^2=0$ picture)
intermediate: Schwarz *Morse Homology* Part II Ch. 1 and Ch. 4; Audin-Damian §4.1; Banyaga-Hurtubise Ch. 7
master: Schwarz *Morse Homology* Part II Ch. 1, 4; Witten 1982; Thom 1949

References

03-modern-geometry/Bott-LecturesOnMorseTheory.pdf · Gradient flows; the Morse complex generated by critical points (background)
Schwarz — Morse Homology (Progress in Mathematics 111, Birkhäuser 1993) · Part II, Ch. 1 and Ch. 4 (the chain group, the boundary operator, and the proof of $\partial^2=0$ from the compactified index-2 moduli)
Witten — Supersymmetry and Morse theory · Journal of Differential Geometry 17 (1982), pp. 661-692 (the deformed differential $d_t = e^{-tf}\,d\,e^{tf}$ and the instanton-counting prediction of the Morse differential)
Thom — Sur une partition en cellules associée à une fonction sur une variété · C.R. Acad. Sci. Paris 228 (1949), pp. 973-975 (the cell decomposition by descending manifolds underlying the complex)
Floer — Witten's complex and infinite-dimensional Morse theory · Journal of Differential Geometry 30 (1989), pp. 207-221 (the moduli-counting realisation of Witten's complex and $\partial^2=0$)
Salamon — Morse theory, the Conley index and Floer homology · Bulletin of the London Mathematical Society 22 (1990), pp. 113-140 (the Morse complex and the boundary identification)

Estimated time

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