File size: 3,104 Bytes
695bd70 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 | 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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