The Morse inequalities for chain complexes of vector spaces

Let K be a chain complex of vector spaces indexed by ℕ, with differentials Kᵢ₊₁ ⟶ Kᵢ, and write cᵢ = dim Kᵢ, bᵢ = dim Hᵢ(K) and qᵢ = dim im (Kᵢ₊₁ ⟶ Kᵢ), with q₋₁ = 0. When Kᵢ is finite-dimensional, rank--nullity gives cᵢ = bᵢ + qᵢ + qᵢ₋₁, and this file draws the three numerical consequences that are known as the Morse inequalities when K is the cellular chain complex of a CW model of a manifold whose cells correspond to the critical points of a Morse function. Since finrank is 0 on infinite-dimensional spaces, each needs finite-dimensional terms.

• Weak inequalities. bᵢ ≤ cᵢ whenever Kᵢ is finite-dimensional, which holds for any homological complex and is proved with the other dimension bounds for complexes of vector spaces (HomologicalComplex.finrank_homology_le_finrank_X).
• Strong inequalities. For every n such that K₀, …, Kₙ are finite-dimensional, bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ cₙ - cₙ₋₁ + ⋯ ± c₀ (ChainComplex.sum_range_neg_one_pow_mul_finrank_homology_le).
• The Morse polynomial identity. If K₀, …, Kₙ are finite-dimensional and Kₙ₊₁ ⟶ Kₙ vanishes, then ∑ᵢ₌₀ⁿ cᵢ tⁱ = ∑ᵢ₌₀ⁿ bᵢ tⁱ + (1 + t) Q(t) in ℕ[t], where Q(t) = ∑ᵢ₌₀ⁿ⁻¹ qᵢ tⁱ records the ranks of the differentials (ChainComplex.sum_range_monomial_finrank_X_eq). Without the vanishing hypothesis the identity holds after adding qₙ tⁿ⁺¹ to the left side (ChainComplex.sum_range_monomial_finrank_X_add_monomial_eq).

The polynomial identity implies the strong inequalities, since the alternating partial sums of the coefficients of (1 + t) Q(t) are the coefficients of Q, which are natural numbers. The weak inequalities and the Euler--Poincaré equality alone do not imply it: the counts c = (2, 0, 0, 2) and b = (1, 0, 0, 1) satisfy both, while c₁ - c₀ < b₁ - b₀.

Conversely, the strong inequalities together with the Euler--Poincaré equality imply the polynomial identity, which is how it is obtained when no chain complex is at hand. For sequences of natural numbers b and c:

• TauCeti.exists_sum_range_monomial_eq_add_mul: if the strong inequalities hold below degree n and the Euler--Poincaré equality holds in degree n, then ∑ᵢ₌₀ⁿ cᵢ tⁱ = ∑ᵢ₌₀ⁿ bᵢ tⁱ + (1 + t) Q(t) with Q ∈ ℕ[t], and the coefficient of tⁱ in Q is the gap (cᵢ - cᵢ₋₁ + ⋯ ± c₀) - (bᵢ - bᵢ₋₁ + ⋯ ± b₀) in the i-th strong inequality.
• TauCeti.eq_of_forall_sum_range_neg_one_pow_mul_le: the lacunary principle. If the strong inequalities hold in every degree and no two consecutive cᵢ are both nonzero, then b = c.

References

• J. Milnor, Morse Theory, Section 5.
• R. Bott, Lectures on Morse theory, old and new, Bulletin of the American Mathematical Society 7 (1982), 331--358, Section 1.

theorem TauCeti.exists_sum_range_monomial_eq_add_mul {b c : ℕ → ℕ} {n : ℕ} (hstrong : ∀ m < n, ∑ i ∈ Finset.range (m + 1), (-1) ^ (m - i) * ↑(b i) ≤ ∑ i ∈ Finset.range (m + 1), (-1) ^ (m - i) * ↑(c i)) (heuler : ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑(b i) = ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑(c i)) : ∃ (Q : Polynomial ℕ), ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (c i) = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (b i) + (1 + Polynomial.X) * Q ∧ (∀ i < n, ↑(Q.coeff i) = ∑ j ∈ Finset.range (i + 1), (-1) ^ (i - j) * ↑(c j) - ∑ j ∈ Finset.range (i + 1), (-1) ^ (i - j) * ↑(b j)) ∧ ∀ (i : ℕ), n ≤ i → Q.coeff i = 0

The Morse polynomial identity from the strong inequalities. Let b and c be sequences of natural numbers satisfying the strong Morse inequalities bₘ - bₘ₋₁ + ⋯ ± b₀ ≤ cₘ - cₘ₋₁ + ⋯ ± c₀ for m < n and the Euler--Poincaré equality ∑ᵢ₌₀ⁿ (-1)ⁱ bᵢ = ∑ᵢ₌₀ⁿ (-1)ⁱ cᵢ. Then ∑ᵢ₌₀ⁿ cᵢ Xⁱ = ∑ᵢ₌₀ⁿ bᵢ Xⁱ + (1 + X) Q for a polynomial Q with natural number coefficients: the coefficient of Xⁱ in Q is the gap (cᵢ - cᵢ₋₁ + ⋯ ± c₀) - (bᵢ - bᵢ₋₁ + ⋯ ± b₀) in the i-th strong inequality for i < n, and zero for i ≥ n.

theorem TauCeti.eq_of_forall_sum_range_neg_one_pow_mul_le {b c : ℕ → ℕ} (hstrong : ∀ (m : ℕ), ∑ i ∈ Finset.range (m + 1), (-1) ^ (m - i) * ↑(b i) ≤ ∑ i ∈ Finset.range (m + 1), (-1) ^ (m - i) * ↑(c i)) (hc : ∀ (i : ℕ), c i = 0 ∨ c (i + 1) = 0) : b = c

The lacunary principle. Let b and c be sequences of natural numbers satisfying the strong Morse inequalities bₘ - bₘ₋₁ + ⋯ ± b₀ ≤ cₘ - cₘ₋₁ + ⋯ ± c₀ in every degree m. If no two consecutive cᵢ are both nonzero, then b = c.

theorem ChainComplex.sum_range_neg_one_pow_mul_finrank_homology_le {k : Type u} [DivisionRing k] (K : ChainComplex (ModuleCat k) ℕ) (n : ℕ) (hfinite : ∀ i ≤ n, Module.Finite k ↑(K.X i)) : ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑(HomologicalComplex.homology K i)) ≤ ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑(K.X i))

The strong Morse inequalities. If the terms of K through degree n are finite-dimensional, then bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ cₙ - cₙ₋₁ + ⋯ ± c₀, where bᵢ is the dimension of Hᵢ(K) and cᵢ that of Kᵢ. The difference of the two sides is the rank of Kₙ₊₁ ⟶ Kₙ.

theorem ChainComplex.sum_range_monomial_finrank_X_add_monomial_eq {k : Type u} [DivisionRing k] (K : ChainComplex (ModuleCat k) ℕ) (n : ℕ) (hfinite : ∀ i ≤ n, Module.Finite k ↑(K.X i)) : ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑(K.X i)) + (Polynomial.monomial (n + 1)) (Module.finrank k ↥(ModuleCat.Hom.hom (K.d (n + 1) n)).range) = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑(HomologicalComplex.homology K i)) + (1 + Polynomial.X) * ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↥(ModuleCat.Hom.hom (K.d (i + 1) i)).range)

The truncated Morse polynomial identity. If the terms of K through degree n are finite-dimensional, then in ℕ[X] ∑ᵢ₌₀ⁿ cᵢ Xⁱ + qₙ Xⁿ⁺¹ = ∑ᵢ₌₀ⁿ bᵢ Xⁱ + (1 + X) ∑ᵢ₌₀ⁿ qᵢ Xⁱ, where cᵢ is the dimension of Kᵢ, bᵢ that of Hᵢ(K) and qᵢ the rank of Kᵢ₊₁ ⟶ Kᵢ.

theorem ChainComplex.sum_range_monomial_finrank_X_eq {k : Type u} [DivisionRing k] (K : ChainComplex (ModuleCat k) ℕ) {n : ℕ} (hfinite : ∀ i ≤ n, Module.Finite k ↑(K.X i)) (hn : K.d (n + 1) n = 0) : ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑(K.X i)) = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑(HomologicalComplex.homology K i)) + (1 + Polynomial.X) * ∑ i ∈ Finset.range n, (Polynomial.monomial i) (Module.finrank k ↥(ModuleCat.Hom.hom (K.d (i + 1) i)).range)

The Morse polynomial identity. If the terms of K through degree n are finite-dimensional and the differential Kₙ₊₁ ⟶ Kₙ vanishes, then in ℕ[X] ∑ᵢ₌₀ⁿ cᵢ Xⁱ = ∑ᵢ₌₀ⁿ bᵢ Xⁱ + (1 + X) Q, where cᵢ is the dimension of Kᵢ, bᵢ that of Hᵢ(K), and Q = ∑ᵢ₌₀ⁿ⁻¹ qᵢ Xⁱ with qᵢ the rank of Kᵢ₊₁ ⟶ Kᵢ: the coefficient of Xⁱ in Q is that rank for i < n and zero otherwise.