Dimensions of relative homology along the long exact sequence of a triple

Let B ⊆ A ⊆ X be a triple of spaces and V a module over a division ring k, and write Hᵢ(-) for relative singular homology with coefficients in V. The long exact sequence of the triple

⋯ ⟶ Hₙ₊₁(X, A) ⟶ Hₙ(A, B) ⟶ Hₙ(X, B) ⟶ Hₙ(X, A) ⟶ ⋯ ⟶ H₀(X, A) ⟶ 0

controls the dimensions of the homology of (X, B) by those of (A, B) and (X, A):

• Hₙ(X, B) is finite-dimensional when Hₙ(A, B) and Hₙ(X, A) are (TauCeti.TopTriple.finite_singularHomology_totalPair), and then dim Hₙ(X, B) ≤ dim Hₙ(A, B) + dim Hₙ(X, A) (TauCeti.TopTriple.finrank_singularHomology_totalPair_le);
• when the homology of (A, B) and (X, A) is finite-dimensional in every degree, the alternating sums Sₙ(P) = dim Hₙ(P) - dim Hₙ₋₁(P) + ⋯ ± dim H₀(P) satisfy Sₙ(A, B) - Sₙ(X, B) + Sₙ(X, A) = dim im (Hₙ₊₁(X, A) ⟶ Hₙ(A, B)) (TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_eq). Hence Sₙ is subadditive, Sₙ(X, B) ≤ Sₙ(A, B) + Sₙ(X, A) (TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_totalPair_le), and when moreover Hₙ₊₁(X, A) vanishes, the alternating sums with signs (-1)ⁱ are additive (TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_totalPair_eq).

Along a filtration A 0 ⊆ A 1 ⊆ ⋯ ⊆ A (N + 1) whose steps have finite-dimensional relative homology in degree i, so does (A (N + 1), A 0) (TauCeti.finite_singularHomology_ofInclusion_of_monotone). When the steps have finite-dimensional relative homology in every degree, the defect ∑ⱼ Sₙ(A (j + 1), A j) - Sₙ(A (N + 1), A 0) of subadditivity is the sum of the ranks of the connecting maps Hₙ₊₁(A (j + 2), A (j + 1)) ⟶ Hₙ(A (j + 1), A 0) of the triples (A (j + 2), A (j + 1), A 0) (TauCeti.sum_range_finrank_range_singularHomologyδ_eq).

Milnor derives the Morse inequalities from these two subadditivity properties, applied to the triples of sublevels of a Morse function. When a compact band is cut at regular values into pieces containing at most one critical value each, every piece satisfies the Morse inequalities with equality, so each coefficient of the polynomial measuring the failure of the Morse inequalities to be equalities is a sum of ranks of connecting maps along the resulting filtration by sublevels. A coarser filtration can hide cancellations within a piece, and then these ranks need not recover the coefficients.

References

• J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Section 5.

theorem TauCeti.TopTriple.finite_singularHomology_totalPair (T : TopTriple) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) (n : ℕ) [Module.Finite k ↑((innerPair.obj T).singularHomology V n)] [Module.Finite k ↑((outerPair.obj T).singularHomology V n)] : Module.Finite k ↑((totalPair.obj T).singularHomology V n)

Finiteness along a triple. If Hₙ(A, B) and Hₙ(X, A) are finite-dimensional, so is Hₙ(X, B).

theorem TauCeti.TopTriple.finrank_singularHomology_totalPair_le (T : TopTriple) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) (n : ℕ) [Module.Finite k ↑((innerPair.obj T).singularHomology V n)] [Module.Finite k ↑((outerPair.obj T).singularHomology V n)] : Module.finrank k ↑((totalPair.obj T).singularHomology V n) ≤ Module.finrank k ↑((innerPair.obj T).singularHomology V n) + Module.finrank k ↑((outerPair.obj T).singularHomology V n)

Subadditivity of dimension along a triple. If Hₙ(A, B) and Hₙ(X, A) are finite-dimensional, then dim Hₙ(X, B) ≤ dim Hₙ(A, B) + dim Hₙ(X, A).

theorem TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_eq (T : TopTriple) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) [∀ (n : ℕ), Module.Finite k ↑((innerPair.obj T).singularHomology V n)] [∀ (n : ℕ), Module.Finite k ↑((outerPair.obj T).singularHomology V n)] (n : ℕ) : ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * (↑(Module.finrank k ↑((innerPair.obj T).singularHomology V i)) - ↑(Module.finrank k ↑((totalPair.obj T).singularHomology V i)) + ↑(Module.finrank k ↑((outerPair.obj T).singularHomology V i))) = ↑(Module.finrank k ↥(ModuleCat.Hom.hom (T.singularHomologyδ V (n + 1) n ⋯)).range)

The dimension count along the long exact sequence of a triple. If the relative homology of (A, B) and of (X, A) is finite-dimensional in every degree, then the alternating sums through degree n satisfy Sₙ(A, B) - Sₙ(X, B) + Sₙ(X, A) = dim im (Hₙ₊₁(X, A) ⟶ Hₙ(A, B)), where Sₙ(P) = dim Hₙ(P) - dim Hₙ₋₁(P) + ⋯ ± dim H₀(P).

theorem TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_totalPair_le (T : TopTriple) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) [∀ (n : ℕ), Module.Finite k ↑((innerPair.obj T).singularHomology V n)] [∀ (n : ℕ), Module.Finite k ↑((outerPair.obj T).singularHomology V n)] (n : ℕ) : ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((totalPair.obj T).singularHomology V i)) ≤ ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((innerPair.obj T).singularHomology V i)) + ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((outerPair.obj T).singularHomology V i))

Subadditivity of the alternating sums along a triple. If the relative homology of (A, B) and of (X, A) is finite-dimensional in every degree, then Sₙ(X, B) ≤ Sₙ(A, B) + Sₙ(X, A), where Sₙ(P) = dim Hₙ(P) - dim Hₙ₋₁(P) + ⋯ ± dim H₀(P).

theorem TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_totalPair_eq (T : TopTriple) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) [∀ (n : ℕ), Module.Finite k ↑((innerPair.obj T).singularHomology V n)] [∀ (n : ℕ), Module.Finite k ↑((outerPair.obj T).singularHomology V n)] (n : ℕ) (hn : CategoryTheory.Limits.IsZero ((outerPair.obj T).singularHomology V (n + 1))) : ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑(Module.finrank k ↑((totalPair.obj T).singularHomology V i)) = ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑(Module.finrank k ↑((innerPair.obj T).singularHomology V i)) + ∑ i ∈ Finset.range (n + 1), (-1) ^ i * ↑(Module.finrank k ↑((outerPair.obj T).singularHomology V i))

Additivity of the alternating sums along a triple. If the relative homology of (A, B) and of (X, A) is finite-dimensional in every degree and Hₙ₊₁(X, A) vanishes, then ∑ᵢ₌₀ⁿ (-1)ⁱ dim Hᵢ(X, B) = ∑ᵢ₌₀ⁿ (-1)ⁱ dim Hᵢ(A, B) + ∑ᵢ₌₀ⁿ (-1)ⁱ dim Hᵢ(X, A).

Finite filtrations

theorem TauCeti.finite_singularHomology_ofInclusion_of_monotone {Y : TopCat} {A : ℕ → Set ↑Y} (hA : Monotone A) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) {N : ℕ} (i : ℕ) (hfin : ∀ j ≤ N, Module.Finite k ↑((TopPair.ofInclusion ⋯).singularHomology V i)) : Module.Finite k ↑((TopPair.ofInclusion ⋯).singularHomology V i)

Finiteness along a filtration. If A 0 ⊆ A 1 ⊆ ⋯ ⊆ A (N + 1) and every step Hᵢ(A (j + 1), A j), j ≤ N, is finite-dimensional, then so is Hᵢ(A (N + 1), A 0).

theorem TauCeti.sum_range_finrank_range_singularHomologyδ_eq {Y : TopCat} {A : ℕ → Set ↑Y} (hA : Monotone A) {k : Type u_1} [DivisionRing k] (V : ModuleCat k) {N : ℕ} (hfin : ∀ j ≤ N, ∀ (i : ℕ), Module.Finite k ↑((TopPair.ofInclusion ⋯).singularHomology V i)) (n : ℕ) : ∑ j ∈ Finset.range N, ↑(Module.finrank k ↥(ModuleCat.Hom.hom ((TopTriple.ofInclusions ⋯ ⋯).singularHomologyδ V (n + 1) n ⋯)).range) = ∑ j ∈ Finset.range (N + 1), ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((TopPair.ofInclusion ⋯).singularHomology V i)) - ∑ i ∈ Finset.range (n + 1), (-1) ^ (n - i) * ↑(Module.finrank k ↑((TopPair.ofInclusion ⋯).singularHomology V i))

The dimension count along a filtration. Let A 0 ⊆ A 1 ⊆ ⋯ ⊆ A (N + 1) be subspaces of a space such that every step Hᵢ(A (j + 1), A j), j ≤ N, is finite-dimensional in every degree, and write Sₙ(P) = dim Hₙ(P) - dim Hₙ₋₁(P) + ⋯ ± dim H₀(P). Then the defect of subadditivity of Sₙ along the filtration, Sₙ(A 1, A 0) + ⋯ + Sₙ(A (N + 1), A N) - Sₙ(A (N + 1), A 0), is the sum over j < N of the ranks of the connecting maps Hₙ₊₁(A (j + 2), A (j + 1)) ⟶ Hₙ(A (j + 1), A 0) of the triples (A (j + 2), A (j + 1), A 0).