| import OAI.Geometry.SingularHomology.SingularSmallChainSupport |
| import OAI.Geometry.SingularHomology.SimplexSingularCompatibility |
| import OAI.Geometry.SingularHomology.IteratedSubdivisionMesh |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Problem340.SingularSmallChains |
|
|
| open SingularSubdivision |
|
|
| variable {X : Type} [TopologicalSpace X] {ι : Type*} |
|
|
| theorem simplexChain_mem_range_inclusion (U : ι → Set X) {n : ℕ} |
| (σ : RealSingularSimplex X n) (hσ : ∃ i, Set.range σ ⊆ U i) : |
| SingularChainFormulas.simplexChain σ ∈ LinearMap.range ((inclusion U).f n).hom := by |
| apply (mem_range_inclusion_iff_supported U n _).mpr |
| change (singularChainsFinsuppEquiv X n).symm (realSingularGenerator σ 1) ∈ _ |
| rw [← singularChainsFinsuppEquiv_single X n σ 1, LinearEquiv.symm_apply_apply] |
| exact Finsupp.single_mem_supported ℝ 1 hσ |
|
|
| theorem pushforward_realize_mem_range_inclusion (U : ι → Set X) (k n : ℕ) |
| (σ : C(StandardSimplex k, X)) (c : Chains (StandardSimplex k) (n + 1)) |
| (hc : ∀ w ∈ c.support, |
| ∃ i, Set.range (σ.comp (AffineSimplex.simplexRealizeMap w)) ⊆ U i) : |
| ((realSingularChainMap σ σ.continuous).f n).hom |
| (SimplexSingularRealization.realize k n c) ∈ |
| LinearMap.range ((inclusion U).f n).hom := by |
| classical |
| rw [SimplexSingularRealization.realize, Finsupp.linearCombination_apply, |
| Finsupp.sum, map_sum] |
| apply (LinearMap.range ((inclusion U).f n).hom).sum_mem |
| intro w hw |
| rw [map_smul, SingularChainFormulas.simplexChain_map] |
| exact (LinearMap.range ((inclusion U).f n).hom).smul_mem _ |
| (simplexChain_mem_range_inclusion U _ (hc w hw)) |
|
|
| theorem exists_iterated_realize_mem_range_inclusion {I : Type} |
| (U : I → Set X) (hopen : ∀ i, IsOpen (U i)) |
| (hcover : (Set.univ : Set X) ⊆ ⋃ i, U i) |
| (n : ℕ) (σ : RealSingularSimplex X n) : |
| ∃ N : ℕ, ∀ k ≥ N, |
| ((realSingularChainMap σ σ.continuous).f n).hom |
| (SimplexSingularRealization.realize n n |
| ((subdivision (simplexCenters n) (n + 1))^[k] |
| (generator (standardVertices n)))) ∈ |
| LinearMap.range ((inclusion U).f n).hom := by |
| classical |
| obtain ⟨N, hN⟩ := exists_iterated_subdivision_small_images n {σ} U hopen hcover |
| refine ⟨N, fun k hk => ?_⟩ |
| apply pushforward_realize_mem_range_inclusion U n n σ |
| exact hN k hk σ (Finset.mem_singleton_self σ) |
|
|
| theorem exists_iterated_realizeInSimplex_mem_range_inclusion {I : Type} |
| (U : I → Set X) (hopen : ∀ i, IsOpen (U i)) |
| (hcover : (Set.univ : Set X) ⊆ ⋃ i, U i) |
| (n : ℕ) (σ : RealSingularSimplex X n) : |
| ∃ N : ℕ, ∀ k ≥ N, |
| ((realSingularChainMap σ σ.continuous).f n).hom |
| (AffineSingularRealization.realizeInSimplex n n |
| ((subdivision (simplexCenters n) (n + 1))^[k] |
| (generator (standardVertices n)))) ∈ |
| LinearMap.range ((inclusion U).f n).hom := by |
| simpa only [SimplexSingularRealization.realize_eq_realizeInSimplex] using |
| exists_iterated_realize_mem_range_inclusion U hopen hcover n σ |
|
|
| end Problem340.SingularSmallChains |
|
|
| end |
|
|
| end OAI |
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|
