The Morse inequalities for CW complexes

Let (C, D) be a relative CW complex with cᵢ relative i-cells, and let bᵢ be the dimension of the relative singular homology Hᵢ(C, D; M) with coefficients in a finite-dimensional module M over a division ring k. Applying the Morse inequalities for chain complexes to the cellular chain complex, and the cellular--singular comparison, gives:

• weak inequalities bᵢ ≤ cᵢ · dim M (TauCeti.finrank_singularHomology_complexBasePair_le);
• strong inequalities bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ (cₙ - cₙ₋₁ + ⋯ ± c₀) · dim M, which need only finitely many cells through degree n (TauCeti.sum_range_neg_one_pow_mul_finrank_singularHomology_complexBasePair_le);
• the Morse polynomial identity ∑ᵢ₌₀ⁿ cᵢ dim M · tⁱ = ∑ᵢ₌₀ⁿ bᵢ tⁱ + (1 + t) Q(t) in ℕ[t] when there are no (n + 1)-cells, where the coefficient of tⁱ in Q is the rank of the cellular differential in degree i + 1 for i < n, and zero otherwise (TauCeti.sum_range_monomial_natCard_cell_mul_finrank_eq).

The same statements for the ordinary singular homology of an absolute CW complex are obtained through the quotient map for the empty subspace. When the cells of a CW model of a manifold correspond to the critical points of a Morse function, with an i-cell for each critical point of index i, these are the Morse inequalities for that function.

References

• J. Milnor, Morse Theory, Section 5.
• A. Hatcher, Algebraic Topology, Section 2.2, Theorem 2.35.

theorem TauCeti.finrank_singularHomology_complexBasePair_le {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] {D : Set X} (C : Set X) [Topology.RelCWComplex C D] (M : ModuleCat k) [Module.Finite k ↑M] (n : ℕ) [Finite (Topology.RelCWComplex.cell C n)] : Module.finrank k ↑((complexBasePair C).singularHomology M n) ≤ Nat.card (Topology.RelCWComplex.cell C n) * Module.finrank k ↑M

The weak Morse inequality for a CW pair. In a degree with finitely many relative cells, the relative singular homology has dimension at most the number of cells times dim M.

theorem TauCeti.sum_range_neg_one_pow_mul_finrank_singularHomology_complexBasePair_le {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] {D : Set X} (C : Set X) [Topology.RelCWComplex C D] (M : ModuleCat k) [Module.Finite k ↑M] {n : ℕ} (hfinite : ∀ i ≤ n, Finite (Topology.RelCWComplex.cell C i)) : ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((complexBasePair C).singularHomology M i)) ≤ ↑(Module.finrank k ↑M) * ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Nat.card (Topology.RelCWComplex.cell C i))

The strong Morse inequalities for a CW pair. If there are finitely many relative cells through degree n, then bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ (cₙ - cₙ₋₁ + ⋯ ± c₀) · dim M, where bᵢ is the dimension of the relative singular homology and cᵢ the number of relative i-cells.

theorem TauCeti.sum_range_monomial_natCard_cell_mul_finrank_eq {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] {D : Set X} (C : Set X) [Topology.RelCWComplex C D] (M : ModuleCat k) [Module.Finite k ↑M] {n : ℕ} (hfinite : ∀ i ≤ n, Finite (Topology.RelCWComplex.cell C i)) (hn : IsEmpty (Topology.RelCWComplex.cell C (n + 1))) : ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Nat.card (Topology.RelCWComplex.cell C i) * Module.finrank k ↑M) = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑((complexBasePair C).singularHomology M i)) + (1 + Polynomial.X) * ∑ i ∈ Finset.range n, (Polynomial.monomial i) (Module.finrank k ↥(ModuleCat.Hom.hom ((cellularChainComplex C M).d (i + 1) i)).range)

The Morse polynomial identity for a CW pair. If there are finitely many relative cells through degree n and none in degree n + 1, then in ℕ[X] ∑ᵢ₌₀ⁿ cᵢ dim M · Xⁱ = ∑ᵢ₌₀ⁿ bᵢ Xⁱ + (1 + X) Q, where cᵢ is the number of relative i-cells, bᵢ the dimension of the relative singular homology, and the coefficient of Xⁱ in Q is the rank of the cellular differential in degree i + 1 for i < n, and zero otherwise.

theorem TauCeti.finrank_singularHomology_le_of_cwComplex {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] (C : Set X) [Topology.CWComplex C] (M : ModuleCat k) [Module.Finite k ↑M] (n : ℕ) [Finite (Topology.RelCWComplex.cell C n)] : Module.finrank k ↑(((AlgebraicTopology.singularHomologyFunctor (ModuleCat k) n).obj M).obj ↧↑C) ≤ Nat.card (Topology.RelCWComplex.cell C n) * Module.finrank k ↑M

The weak Morse inequality for a CW complex. In a degree with finitely many cells, the singular homology has dimension at most the number of cells times dim M.

theorem TauCeti.sum_range_neg_one_pow_mul_finrank_singularHomology_le_of_cwComplex {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] (C : Set X) [Topology.CWComplex C] (M : ModuleCat k) [Module.Finite k ↑M] {n : ℕ} (hfinite : ∀ i ≤ n, Finite (Topology.RelCWComplex.cell C i)) : ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑(((AlgebraicTopology.singularHomologyFunctor (ModuleCat k) i).obj M).obj ↧↑C)) ≤ ↑(Module.finrank k ↑M) * ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Nat.card (Topology.RelCWComplex.cell C i))

The strong Morse inequalities for a CW complex. If there are finitely many cells through degree n, then bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ (cₙ - cₙ₋₁ + ⋯ ± c₀) · dim M, where bᵢ is the dimension of the singular homology and cᵢ the number of i-cells.

theorem TauCeti.sum_range_monomial_natCard_cell_mul_finrank_eq_of_cwComplex {X : Type w} [TopologicalSpace X] [T2Space X] {k : Type w} [DivisionRing k] (C : Set X) [Topology.CWComplex C] (M : ModuleCat k) [Module.Finite k ↑M] {n : ℕ} (hfinite : ∀ i ≤ n, Finite (Topology.RelCWComplex.cell C i)) (hn : IsEmpty (Topology.RelCWComplex.cell C (n + 1))) : ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Nat.card (Topology.RelCWComplex.cell C i) * Module.finrank k ↑M) = ∑ i ∈ Finset.range (n + 1), (Polynomial.monomial i) (Module.finrank k ↑(((AlgebraicTopology.singularHomologyFunctor (ModuleCat k) i).obj M).obj ↧↑C)) + (1 + Polynomial.X) * ∑ i ∈ Finset.range n, (Polynomial.monomial i) (Module.finrank k ↥(ModuleCat.Hom.hom ((cellularChainComplex C M).d (i + 1) i)).range)

The Morse polynomial identity for a CW complex. If there are finitely many cells through degree n and none in degree n + 1, then in ℕ[X] ∑ᵢ₌₀ⁿ cᵢ dim M · Xⁱ = ∑ᵢ₌₀ⁿ bᵢ Xⁱ + (1 + X) Q, where cᵢ is the number of i-cells, bᵢ the dimension of the singular homology, and the coefficient of Xⁱ in Q is the rank of the cellular differential in degree i + 1 for i < n, and zero otherwise.