Gluing parallel sections of local homology

On a finite-dimensional Hausdorff manifold without boundary, a family of local homology classes invariant under path transport is locally represented by a single relative class. In the manifold dimension, it therefore glues uniquely over every compact subset. For a compact manifold, this gives a unique absolute homology class with the prescribed local values. For integral coefficients and a parallel family of local generators, this is the existence and uniqueness argument for the fundamental class.

No connectedness assumption is needed: orientations may be chosen independently on components. The comparison of homological orientations with tangent-space orientations, and the boundary formula for manifolds with boundary, are separate matters.

References

• A. Hatcher, Algebraic Topology, Section 3.3, Lemma 3.27 and Theorem 3.26.

theorem TauCeti.exists_mem_nhds_localHomologyRestriction_eq_of_invariant {Λ : Type w} [Ring Λ] (N : ModuleCat Λ) {M E : Type w} [TopologicalSpace M] [T2Space M] [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [ChartedSpace E M] (k : ℕ) (α : (x : M) → ↑(localHomology N {x} k)) (hα : ∀ {x y : FundamentalGroupoid M} (f : x ⟶ y), (CategoryTheory.ConcreteCategory.hom (localHomologyTransport N M E k f)) (α x.as) = α y.as) (x : M) : ∃ U ∈ nhds x, ∃ (β : ↑(localHomology N U k)), ∀ (y : M) (hy : y ∈ U), (CategoryTheory.ConcreteCategory.hom (localHomologyRestriction N ⋯ k)) β = α y

A transport-invariant family of local homology classes is represented near every point by a single relative class. This holds in every degree, not only in the dimension of the manifold.

theorem TauCeti.existsUnique_localHomologyRestriction_eq_of_invariant {Λ : Type w} [Ring Λ] (N : ModuleCat Λ) {M E : Type w} [TopologicalSpace M] [T2Space M] [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [ChartedSpace E M] {n : ℕ} (hn : Module.finrank ℝ E = n) {A : Set M} (hA : IsCompact A) (α : (x : M) → ↑(localHomology N {x} n)) (hα : ∀ {x y : FundamentalGroupoid M} (f : x ⟶ y), (CategoryTheory.ConcreteCategory.hom (localHomologyTransport N M E n f)) (α x.as) = α y.as) : ∃! γ : ↑(localHomology N A n), ∀ (x : M) (hx : x ∈ A), (CategoryTheory.ConcreteCategory.hom (localHomologyRestriction N ⋯ n)) γ = α x

A parallel family of top-dimensional local classes glues uniquely over a compact subset. The compatibility hypothesis is path transport, rather than the existence of local lifts.

theorem TauCeti.existsUnique_singularHomology_localHomologyRestriction_eq_of_invariant {Λ : Type w} [Ring Λ] (N : ModuleCat Λ) {M E : Type w} [TopologicalSpace M] [T2Space M] [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimensional ℝ E] [ChartedSpace E M] [CompactSpace M] {n : ℕ} (hn : Module.finrank ℝ E = n) (α : (x : M) → ↑(localHomology N {x} n)) (hα : ∀ {x y : FundamentalGroupoid M} (f : x ⟶ y), (CategoryTheory.ConcreteCategory.hom (localHomologyTransport N M E n f)) (α x.as) = α y.as) : ∃! γ : ↑((TopCat.toSSet.obj ↧M).homology N n), ∀ (x : M), (CategoryTheory.ConcreteCategory.hom (localHomologyRestriction N ⋯ n)) ((ModuleCat.Hom.hom ((TopPair.ofSubset Set.univᶜ).singularHomologyπ N n)) γ) = α x

On a compact manifold, a parallel family of top-dimensional local classes is the family of local values of exactly one absolute singular homology class. The map to local homology is the ordinary-to-relative quotient map followed by restriction.