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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 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 | import OAI.Geometry.SingularHomology.SimplexCoordinates
import OAI.Geometry.SingularHomology.AffineSimplexGeometry
import OAI.Geometry.SingularHomology.SingularChainsFinsuppBoundary
import OAI.Geometry.SingularHomology.BarycentricSubdivision
namespace OAI
noncomputable section
namespace Problem340.SimplexSingularRealization
open SingularSubdivision
abbrev Simplex (k : ℕ) := Convexity.StdSimplex ℝ (Fin (k + 1))
def realize (k n : ℕ) :
Chains (Simplex k) (n + 1) →ₗ[ℝ] (realSingularChainComplex (Simplex k)).X n :=
Finsupp.linearCombination ℝ (fun v =>
SingularChainFormulas.simplexChain (AffineSimplex.simplexRealizeMap v))
@[simp] theorem realize_generator (k n : ℕ) (v : Fin (n + 1) → Simplex k) :
realize k n (generator v) =
SingularChainFormulas.simplexChain (AffineSimplex.simplexRealizeMap v) := by
simp [realize, generator]
theorem face_simplexRealizeMap (k n : ℕ) (v : Fin (n + 2) → Simplex k)
(i : Fin (n + 2)) :
SingularChainFormulas.face (AffineSimplex.simplexRealizeMap v) i =
AffineSimplex.simplexRealizeMap (face i v) :=
AffineSimplex.simplexRealizeMap_comp_map i.succAbove v
theorem boundary_realize (k n : ℕ) (c : Chains (Simplex k) (n + 2)) :
((realSingularChainComplex (Simplex k)).d (n + 1) n).hom (realize k (n + 1) c) =
realize k n (boundary (n + 1) c) := by
have h : ((realSingularChainComplex (Simplex k)).d (n + 1) n).hom.comp
(realize k (n + 1)) = (realize k n).comp (boundary (n + 1)) := by
apply linearMap_ext
intro v
change ((realSingularChainComplex (Simplex k)).d (n + 1) n).hom
(realize k (n + 1) (generator v)) = realize k n (boundary (n + 1) (generator v))
rw [realize_generator, SingularChainFormulas.simplexChain_boundary_real,
boundary_generator, map_sum]
apply Finset.sum_congr rfl
intro i _
rw [map_smul, realize_generator, face_simplexRealizeMap]
exact LinearMap.congr_fun h c
theorem affineMap_realize (k l n : ℕ) (w : Fin (k + 1) → Simplex l)
(c : Chains (Simplex k) (n + 1)) :
((realSingularChainMap (AffineSimplex.simplexRealizeMap w)
(AffineSimplex.simplexRealizeMap w).continuous).f n).hom (realize k n c) =
realize l n (mapVertices (AffineSimplex.simplexRealizeMap w) (n + 1) c) := by
have h : ((realSingularChainMap (AffineSimplex.simplexRealizeMap w)
(AffineSimplex.simplexRealizeMap w).continuous).f n).hom.comp (realize k n) =
(realize l n).comp (mapVertices (AffineSimplex.simplexRealizeMap w) (n + 1)) := by
apply linearMap_ext
intro v
change ((realSingularChainMap (AffineSimplex.simplexRealizeMap w)
(AffineSimplex.simplexRealizeMap w).continuous).f n).hom
(realize k n (generator v)) = _
rw [realize_generator, SingularChainFormulas.simplexChain_map]
have he : (AffineSimplex.simplexRealizeMap w).comp
(AffineSimplex.simplexRealizeMap v) =
AffineSimplex.simplexRealizeMap (AffineSimplex.simplexRealizeMap w ∘ v) := by
ext p : 1
exact AffineSimplex.simplexRealizeMap_assoc w v p
simp only [LinearMap.comp_apply, mapVertices_generator, realize_generator]
rw [he]
exact LinearMap.congr_fun h c
theorem map_realize (k l n : ℕ) (f : Fin (k + 1) → Fin (l + 1))
(c : Chains (Simplex k) (n + 1)) :
((realSingularChainMap (Convexity.StdSimplex.map f) (Convexity.StdSimplex.continuous_map ℝ f)).f n).hom
(realize k n c) =
realize l n (mapVertices (Convexity.StdSimplex.map f) (n + 1) c) := by
let F : C(Simplex k, Simplex l) := ⟨Convexity.StdSimplex.map f, Convexity.StdSimplex.continuous_map ℝ f⟩
have h : ((realSingularChainMap F F.continuous).f n).hom.comp (realize k n) =
(realize l n).comp (mapVertices (Convexity.StdSimplex.map f) (n + 1)) := by
apply linearMap_ext
intro v
change ((realSingularChainMap F F.continuous).f n).hom
(realize k n (generator v)) = _
rw [realize_generator, SingularChainFormulas.simplexChain_map]
have he : F.comp (AffineSimplex.simplexRealizeMap v) =
AffineSimplex.simplexRealizeMap (Convexity.StdSimplex.map f ∘ v) := by
ext p : 1
exact AffineSimplex.map_simplexRealizeMap f v p
simp only [LinearMap.comp_apply, mapVertices_generator, realize_generator]
rw [he]
exact LinearMap.congr_fun h c
@[simp] theorem simplexRealizeMap_standardVertices (n : ℕ) :
AffineSimplex.simplexRealizeMap (standardVertices n) = ContinuousMap.id (Simplex n) := by
ext p : 1
apply RealSimplex.coordinates_injective
funext j
change AffineSimplex.simplexRealizeMap (standardVertices n) p j = p j
rw [AffineSimplex.simplexRealizeMap_apply]
simp [standardVertices, Finsupp.single_apply]
@[simp] theorem realize_standardVertices (n : ℕ) :
realize n n (generator (standardVertices n)) =
SingularChainFormulas.simplexChain (ContinuousMap.id (Simplex n)) := by
rw [realize_generator, simplexRealizeMap_standardVertices]
end Problem340.SimplexSingularRealization
end
end OAI
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