Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Algebra.GroupRing.HomotopyFunctor

namespace OAI

noncomputable section

open Set Metric Topology BigOperators
open Set Metric Topology

namespace GroupRingTopology.Cellular
universe u
variable {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] [Nonempty X]
variable {C : Set X} [CWComplex C] {f : X → Y} (hf : IsClosedEmbedding f)

def transportedMap (n : ℕ) (i : RelCWComplex.cell C n) : PartialEquiv (Fin n → ℝ) Y :=
  (RelCWComplex.map n i).trans (hf.isEmbedding.toPartialHomeomorph f).toPartialEquiv

@[simp] lemma transportedMap_apply (n : ℕ) (i : RelCWComplex.cell C n) (x : Fin n → ℝ) :
    transportedMap hf n i x = f (RelCWComplex.map n i x) := rfl

lemma transportedMap_source (n : ℕ) (i : RelCWComplex.cell C n) :
    (transportedMap hf n i).source = ball 0 1 := by
  simp only [transportedMap,PartialEquiv.trans_source,
    IsEmbedding.toPartialHomeomorph_source,Set.preimage_univ,Set.inter_univ,
    RelCWComplex.source_eq]

lemma transportedMap_image (n : ℕ) (i : RelCWComplex.cell C n) (S : Set (Fin n → ℝ)) :
    transportedMap hf n i '' S = f '' (RelCWComplex.map n i '' S) :=
  Set.image_comp f (RelCWComplex.map n i) S

lemma transportedMap_continuousOn (n : ℕ) (i : RelCWComplex.cell C n) :
    ContinuousOn (transportedMap hf n i) (closedBall 0 1) :=
  hf.continuous.comp_continuousOn (RelCWComplex.continuousOn n i)

lemma transportedMap_continuousOn_symm (n : ℕ) (i : RelCWComplex.cell C n) :
    ContinuousOn (transportedMap hf n i).symm (transportedMap hf n i).target := by
  let e := hf.isEmbedding.toPartialHomeomorph f
  change ContinuousOn (fun y => (RelCWComplex.map n i).symm (e.symm y))
    (e.target ∩ e.symm ⁻¹' (RelCWComplex.map n i).target)
  exact (RelCWComplex.continuousOn_symm n i).comp
    (e.continuousOn_symm.mono Set.inter_subset_left) (fun _ h => h.2)

variable [T2Space X]

/-- Closed embeddings transport classical CW structures, including their weak
 topology, with precisely the same cells. -/
@[instance_reducible]
def transportCW : CWComplex (f '' C) where
  cell := RelCWComplex.cell C
  map := transportedMap hf
  source_eq := transportedMap_source hf
  continuousOn := transportedMap_continuousOn hf
  continuousOn_symm := transportedMap_continuousOn_symm hf
  pairwiseDisjoint' := by
    intro p _ q _ hpq
    change Disjoint (transportedMap hf p.1 p.2 '' ball 0 1)
      (transportedMap hf q.1 q.2 '' ball 0 1)
    rw [transportedMap_image,transportedMap_image]
    exact (CWComplex.disjoint_openCell_of_ne hpq).image hf.injective.injOn (Set.subset_univ _) (Set.subset_univ _)
  mapsTo' := by
    intro n i
    obtain ⟨I,hI⟩ := CWComplex.cellFrontier_subset_finite_closedCell n i
    refine ⟨I,?_⟩
    intro x hx
    have h := hI (show RelCWComplex.map n i x ∈ CWComplex.cellFrontier n i from ⟨x,hx,rfl⟩)
    simp only [Set.mem_iUnion] at h
    obtain ⟨m,hm,j,hj,hx⟩ := h
    refine Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨hm,Set.mem_iUnion.mpr
      ⟨j,Set.mem_iUnion.mpr ⟨hj,?_⟩⟩⟩⟩
    rw [transportedMap_image]
    exact ⟨_,hx,rfl⟩
  closed' := by
    intro A hA hclosed
    have hpre : IsClosed (f ⁻¹' A) := by
      apply (CWComplex.closed C (f ⁻¹' A) ?_).mpr
      · intro n i
        have h := (hclosed n i).preimage hf.continuous
        rw [transportedMap_image,Set.preimage_inter,
          Set.preimage_image_eq _ hf.injective] at h
        exact h
      · intro x hx
        obtain ⟨y,hy,hyx⟩ := hA hx
        exact hf.injective hyx ▸ hy
    have h := hf.isClosedMap _ hpre
    rwa [Set.image_preimage_eq_of_subset (hA.trans (Set.image_subset_range f C))] at h
  union' := by
    simp only [transportedMap_image,← Set.image_iUnion]
    change f '' (⋃ n, ⋃ i, CWComplex.closedCell n i) = f '' C
    rw [CWComplex.union]

end GroupRingTopology.Cellular
open Set Metric Topology
namespace GroupRingTopology.Attachment

section
open Set Metric Topology
universe z
variable {X : Type z} [TopologicalSpace X] [T2Space X] [Nonempty X]
variable [CWComplex (Set.univ : Set X)] {ι : Type z} {n : ℕ}
variable (a : ι → C(Sphere n,X))

abbrev OldCell (m : ℕ) := RelCWComplex.cell (Set.univ : Set X) m

omit [Nonempty X] in
lemma attaching_finite_frontier
    (hdim : ∀ m (_ : OldCell (X := X) m), m < n) (i : ι) :
    ∃ I : ∀ m, Finset (OldCell (X := X) m),
      ∀ x, a i x ∈ ⋃ m < n, ⋃ j ∈ I m, CWComplex.closedCell m j := by
  classical
  let S := {p : Σ m, OldCell (X := X) m |
    (Set.range (a i) ∩ CWComplex.openCell p.1 p.2).Nonempty}
  have hS : S.Finite := Cellular.compact_finitely_many_cells (C := (Set.univ : Set X))
    (isCompact_range (a i).continuous)
  have hSm (m : ℕ) : ((Sigma.mk m) ⁻¹' S).Finite :=
    hS.preimage (Set.injOn_of_injective (fun _ _ h => eq_of_heq (Sigma.mk.inj_iff.mp h).2))
  refine ⟨fun m => (hSm m).toFinset,?_⟩
  intro x
  have hx : a i x ∈ (Set.univ : Set X) := Set.mem_univ _
  rw [← CWComplex.iUnion_openCell_eq_complex (C := (Set.univ : Set X))] at hx
  simp only [Set.mem_iUnion] at hx
  obtain ⟨m,j,hx⟩ := hx
  exact Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨hdim m j,Set.mem_iUnion.mpr
    ⟨j,Set.mem_iUnion.mpr ⟨(hSm m).mem_toFinset.mpr ⟨a i x,⟨x,rfl⟩,hx⟩,
      CWComplex.openCell_subset_closedCell m j hx⟩⟩⟩⟩

abbrev LayerCell (m : ℕ) := OldCell (X := X) m ⊕ CellIndex.{z,z} m n ι

def layerMap (m : ℕ) : LayerCell (X := X) (ι := ι) (n := n) m →
    PartialEquiv (Fin m → ℝ) (Space a)
  | Sum.inl i => Cellular.transportedMap (C := Set.univ) (base_closedEmbedding a) m i
  | Sum.inr i => cellMap a m i

omit [T2Space X] in
lemma layerMap_old_apply (m : ℕ) (i : OldCell (X := X) m) (x : Fin m → ℝ) :
    layerMap a m (Sum.inl i) x = base a (RelCWComplex.map m i x) := rfl

omit [T2Space X] in
lemma layerMap_old_image (m : ℕ) (i : OldCell (X := X) m) (S : Set (Fin m → ℝ)) :
    layerMap a m (Sum.inl i) '' S = base a '' (RelCWComplex.map m i '' S) :=
  Cellular.transportedMap_image (base_closedEmbedding a) m i S

omit [T2Space X] in
lemma layerMap_new (i : ι) :
    layerMap a n (Sum.inr ⟨⟨⟨rfl⟩,i⟩⟩) = characteristicPE a i := rfl

@[instance_reducible]
def layerCW (hdim : ∀ m (_ : OldCell (X := X) m), m < n) :
    CWComplex (Set.univ : Set (Space a)) where
  cell := LayerCell (X := X) (ι := ι) (n := n)
  map := layerMap a
  source_eq := by
    intro m i
    cases i with
    | inl i => exact Cellular.transportedMap_source (base_closedEmbedding a) m i
    | inr i => exact (relCWComplex a).source_eq m i
  continuousOn := by
    intro m i
    cases i with
    | inl i => exact Cellular.transportedMap_continuousOn (base_closedEmbedding a) m i
    | inr i => exact (relCWComplex a).continuousOn m i
  continuousOn_symm := by
    intro m i
    cases i with
    | inl i => exact Cellular.transportedMap_continuousOn_symm (base_closedEmbedding a) m i
    | inr i => exact (relCWComplex a).continuousOn_symm m i
  pairwiseDisjoint' := by
    rintro ⟨m,i⟩ _ ⟨k,j⟩ _ hne
    change Disjoint (layerMap a m i '' ball 0 1) (layerMap a k j '' ball 0 1)
    cases i with
    | inl i =>
      cases j with
      | inl j =>
        rw [layerMap_old_image,layerMap_old_image]
        apply (CWComplex.disjoint_openCell_of_ne ?_).image (base_closedEmbedding a).injective.injOn (Set.subset_univ _) (Set.subset_univ _)
        intro h
        cases h
        exact hne rfl
      | inr j =>
        obtain ⟨⟨⟨hk⟩,j⟩⟩ := j
        subst k
        rw [layerMap_old_image,layerMap_new,characteristicPE_image_ball]
        apply Set.disjoint_left.mpr
        rintro z ⟨x,_,rfl⟩ ⟨y,h⟩
        cases h
    | inr i =>
      obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
      subst m
      cases j with
      | inl j =>
        rw [layerMap_new,characteristicPE_image_ball,layerMap_old_image]
        apply Set.disjoint_left.mpr
        rintro z ⟨x,rfl⟩ ⟨y,_,h⟩
        cases h
      | inr j =>
        obtain ⟨⟨⟨hk⟩,j⟩⟩ := j
        subst k
        rw [layerMap_new,layerMap_new,characteristicPE_image_ball,characteristicPE_image_ball]
        apply Set.disjoint_left.mpr
        rintro z ⟨x,rfl⟩ ⟨y,h⟩
        have hij : j = i := congrArg Sigma.fst (Sum.inr_injective h)
        subst j
        exact hne rfl
  mapsTo' := by
    classical
    intro m c
    cases c with
    | inl i =>
      obtain ⟨I,hI⟩ := CWComplex.cellFrontier_subset_finite_closedCell m i
      refine ⟨fun k => (I k).image Sum.inl,?_⟩
      intro x hx
      have hh := hI (show RelCWComplex.map m i x ∈ CWComplex.cellFrontier m i from ⟨x,hx,rfl⟩)
      simp only [Set.mem_iUnion] at hh
      obtain ⟨k,hkm,j,hj,hx⟩ := hh
      refine Set.mem_iUnion.mpr ⟨k,Set.mem_iUnion.mpr ⟨hkm,Set.mem_iUnion.mpr
        ⟨Sum.inl j,Set.mem_iUnion.mpr ⟨Finset.mem_image.mpr ⟨j,hj,rfl⟩,?_⟩⟩⟩⟩
      rw [layerMap_old_image]
      exact ⟨_,hx,rfl⟩
    | inr i =>
      obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
      subst m
      obtain ⟨I,hI⟩ := attaching_finite_frontier a hdim i
      refine ⟨fun k => (I k).image Sum.inl,?_⟩
      intro x hx
      have hh := hI ⟨x,hx⟩
      simp only [Set.mem_iUnion] at hh
      obtain ⟨k,hkn,j,hj,hx'⟩ := hh
      refine Set.mem_iUnion.mpr ⟨k,Set.mem_iUnion.mpr ⟨hkn,Set.mem_iUnion.mpr
        ⟨Sum.inl j,Set.mem_iUnion.mpr ⟨Finset.mem_image.mpr ⟨j,hj,rfl⟩,?_⟩⟩⟩⟩
      rw [layerMap_old_image]
      refine ⟨a i ⟨x,hx⟩,hx',?_⟩
      change base a (a i ⟨x,hx⟩) = characteristic a i x
      exact (cell_boundary a i ⟨x,hx⟩).symm.trans
        (characteristic_disk a i (boundary n ⟨x,hx⟩)).symm
  closed' := by
    intro A _ hA
    apply (isClosed_iff a A).mpr
    constructor
    · apply (CWComplex.closed Set.univ (base a ⁻¹' A) (Set.subset_univ _)).mpr
      intro m i
      have h := (hA m (Sum.inl i)).preimage (continuous_base a)
      rw [layerMap_old_image,Set.preimage_inter,
        Set.preimage_image_eq _ (base_closedEmbedding a).injective] at h
      exact h
    · intro i
      have h := (hA n (Sum.inr ⟨⟨⟨rfl⟩,i⟩⟩)).preimage (cell a i).continuous
      rw [layerMap_new,characteristicPE_image_closedBall,Set.preimage_inter] at h
      simpa only [Set.preimage_range,Set.inter_univ] using h
  union' := by
    apply Set.eq_univ_of_forall
    intro z
    cases z with
    | inl x =>
      have hx : x ∈ (Set.univ : Set X) := Set.mem_univ _
      rw [← CWComplex.union (C := (Set.univ : Set X))] at hx
      simp only [Set.mem_iUnion] at hx
      obtain ⟨m,i,hx⟩ := hx
      refine Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨Sum.inl i,?_⟩⟩
      rw [layerMap_old_image]
      exact ⟨x,hx,rfl⟩
    | inr p =>
      refine Set.mem_iUnion.mpr ⟨n,Set.mem_iUnion.mpr ⟨Sum.inr ⟨⟨⟨rfl⟩,p.1⟩⟩,?_⟩⟩
      rw [layerMap_new,characteristicPE_image_closedBall]
      exact ⟨interior n p.2,cell_interior a p.1 p.2⟩

lemma layerCW_dimensions (hdim : ∀ m (_ : OldCell (X := X) m), m < n)
    (m : ℕ) (i : (layerCW a hdim).cell m) : m < n+1 := by
  cases i with
  | inl i => exact (hdim m i).trans (Nat.lt_succ_self n)
  | inr i =>
    obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
    subst m
    exact Nat.lt_succ_self n

lemma layerCW_finite_cells [Finite ι]
    (hfin : ∀ m, Finite (OldCell (X := X) m))
    (hdim : ∀ m (_ : OldCell (X := X) m), m < n) (m : ℕ) :
    Finite ((layerCW a hdim).cell m) := by
  let := hfin m
  change Finite (LayerCell (X := X) (ι := ι) (n := n) m)
  infer_instance

end

section
open Set Metric Topology
universe u v w z
variable {X : Type u} [TopologicalSpace X] {ι : Type v} {n : ℕ}
variable (a : ι → C(Sphere n,X))
variable {G : Type w} [Group G] [MulAction G X] [MulAction G ι]

instance spaceMulAction : MulAction G (Space a) where
  smul g := Sum.elim (fun x => Sum.inl (g • x)) (fun p => Sum.inr ⟨g • p.1,p.2⟩)
  one_smul := by
    intro x
    cases x with
    | inl x => change Sum.inl ((1 : G) • x) = Sum.inl x; rw [one_smul]
    | inr p => change Sum.inr ⟨(1 : G) • p.1,p.2⟩ = Sum.inr p; rw [one_smul]
  mul_smul := by
    intro g h x
    cases x with
    | inl x => change Sum.inl ((g * h) • x) = Sum.inl (g • h • x); rw [mul_smul]
    | inr p => change (Sum.inr ⟨(g * h) • p.1,p.2⟩ : Space a) = Sum.inr ⟨g • h • p.1,p.2⟩; rw [mul_smul]
@[simp] lemma smul_base (g : G) (x : X) : g • base a x = base a (g • x) := rfl
@[simp] lemma smul_openCell (g : G) (i : ι) (x : OpenDisk n) :
    g • openCell a i x = openCell a (g • i) x := rfl

variable (ha : ∀ (g : G) i x, a (g • i) x = g • a i x)
include ha
lemma smul_cell (g : G) (i : ι) (d : Disk n) :
    g • cell a i d = cell a (g • i) d := by
  classical
  by_cases hd : d.1 ∈ ball (0 : Fin n → ℝ) 1
  · have heq : d = interior n ⟨d.1,hd⟩ := rfl
    rw [heq,cell_interior,cell_interior]
    rfl
  · let x : Sphere n := ⟨d.1,by
      have hm := d.2
      simp only [mem_closedBall,mem_ball,mem_sphere] at *
      exact le_antisymm hm (le_of_not_gt hd)⟩
    have heq : d = boundary n x := rfl
    rw [heq,cell_boundary,cell_boundary,smul_base,ha]

lemma continuousConstSMul [ContinuousConstSMul G X] : ContinuousConstSMul G (Space a) where
  continuous_const_smul g := (continuous_iff a _).mpr ⟨
    (continuous_base a).comp (continuous_const_smul g),fun i => by
      have heq : (g • ·) ∘ cell a i = cell a (g • i) := funext (smul_cell a ha g i)
      rw [heq]
      exact (cell a (g • i)).continuous⟩

omit ha in
lemma isOpen_iff (S : Set (Space a)) :
    IsOpen S ↔ IsOpen (base a ⁻¹' S) ∧ ∀ i, IsOpen (cell a i ⁻¹' S) := by
  change @IsOpen _ (TopologicalSpace.coinduced (quotient a) inferInstance) S ↔ _
  rw [isOpen_coinduced,isOpen_sum_iff,isOpen_sigma_iff]
  rfl

end

open Set Metric Topology
universe u v w
abbrev Collar (n : ℕ) := {d : Disk n | (1 / 2 : ℝ) < ‖d.1‖}

def collarInclusion (n : ℕ) : Collar n → Disk n := Subtype.val

def radial (n : ℕ) : Collar n → Sphere n := fun d =>
  ⟨‖d.1.1‖⁻¹ • d.1.1,by
    have hn : ‖d.1.1‖ ≠ 0 := ne_of_gt (lt_trans (by norm_num) d.2)
    rw [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,
      inv_mul_cancel₀ hn]⟩

lemma continuous_radial (n : ℕ) : Continuous (radial n) := by
  apply Continuous.subtype_mk
  have hn : ∀ d : Collar n, ‖d.1.1‖ ≠ 0 := fun d => ne_of_gt (lt_trans (by norm_num) d.2)
  exact ((continuous_subtype_val.comp continuous_subtype_val).norm.inv₀ hn).smul
    (continuous_subtype_val.comp continuous_subtype_val)

lemma collar_openEmbedding (n : ℕ) : IsOpenEmbedding (collarInclusion n) :=
  (isOpen_lt continuous_const continuous_subtype_val.norm).isOpenEmbedding_subtypeVal

lemma radial_boundary (n : ℕ) (x : Sphere n) (h : (1/2:ℝ) < ‖(boundary n x).1‖) :
    radial n ⟨boundary n x,h⟩ = x := by
  apply Subtype.ext
  have hx : ‖x.1‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using x.2
  simp only [radial,boundary,hx,inv_one,one_smul]

variable {X : Type u} [TopologicalSpace X] {ι : Type v} {n : ℕ}
variable (a : ι → C(Sphere n,X))

/-- An open old-space neighborhood thickened by the outer half of each disk.
Membership is tested by radial retraction of the disk to its attaching sphere. -/
def thickening (U : Set X) : Set (Space a) := fun z =>
  match z with
  | Sum.inl x => x ∈ U
  | Sum.inr p => ∃ h : (1/2:ℝ) < ‖p.2.1‖,
      a p.1 (radial n ⟨interior n p.2,h⟩) ∈ U

@[simp] lemma base_mem_thickening (U : Set X) (x : X) :
    base a x ∈ thickening a U ↔ x ∈ U := Iff.rfl

lemma cell_preimage_thickening (U : Set X) (i : ι) :
    cell a i ⁻¹' thickening a U =
      collarInclusion n '' ((a i ∘ radial n) ⁻¹' U) := by
  classical
  ext d
  by_cases hd : d.1 ∈ ball (0 : Fin n → ℝ) 1
  · have heq : d = interior n ⟨d.1,hd⟩ := rfl
    rw [heq]
    simp only [Set.mem_preimage,cell_interior,Set.mem_image]
    constructor
    · rintro ⟨h,hh⟩
      exact ⟨⟨d,h⟩,hh,rfl⟩
    · rintro ⟨y,hy,he⟩
      have heq' : y.1 = d := he
      subst d
      exact ⟨y.2,hy⟩
  · let x : Sphere n := ⟨d.1,by
      have hm := d.2
      simp only [mem_closedBall,mem_ball,mem_sphere] at *
      exact le_antisymm hm (le_of_not_gt hd)⟩
    have heq : d = boundary n x := rfl
    have hx : ‖x.1‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using x.2
    have hhalf : (1/2:ℝ) < ‖(boundary n x).1‖ := by change (1/2:ℝ) < ‖x.1‖; rw [hx]; norm_num
    rw [heq]
    simp only [Set.mem_preimage,cell_boundary,base_mem_thickening,Set.mem_image]
    constructor
    · intro h
      exact ⟨⟨boundary n x,hhalf⟩,by simpa only [Function.comp_apply,radial_boundary] using h,rfl⟩
    · rintro ⟨y,hy,he⟩
      have heq' : y = ⟨boundary n x,hhalf⟩ := Subtype.ext he
      simpa only [heq',Function.comp_apply,radial_boundary] using hy

lemma thickening_isOpen {U : Set X} (hU : IsOpen U) : IsOpen (thickening a U) := by
  rw [isOpen_iff]
  refine ⟨hU,fun i => ?_⟩
  rw [cell_preimage_thickening]
  exact (collar_openEmbedding n).isOpenMap _
    (hU.preimage ((a i).continuous.comp (continuous_radial n)))

variable {G : Type w} [Group G] [MulAction G X] [MulAction G ι]
variable (ha : ∀ (g : G) i x, a (g • i) x = g • a i x)
include ha
lemma thickening_disjoint {U : Set X}
    (hU : ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1) :
    ∀ g : G, ((g • ·) '' thickening a U ∩ thickening a U).Nonempty → g = 1 := by
  intro g h
  obtain ⟨_,⟨z,hz,rfl⟩,hgz⟩ := h
  cases z with
  | inl x => exact hU g ⟨g • x,⟨x,hz,rfl⟩,hgz⟩
  | inr p =>
    obtain ⟨hhalf,hz⟩ := hz
    obtain ⟨_,hgz⟩ := hgz
    change a (g • p.1) (radial n ⟨interior n p.2,hhalf⟩) ∈ U at hgz
    rw [ha] at hgz
    exact hU g ⟨g • a p.1 (radial n ⟨interior n p.2,hhalf⟩),⟨_,hz,rfl⟩,hgz⟩

lemma exists_disjoint_nhds [IsCancelSMul G ι]
    (hX : ∀ x : X, ∃ U ∈ 𝓝 x, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1) :
    ∀ z : Space a, ∃ U ∈ 𝓝 z, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1 := by
  intro z
  cases z with
  | inl x =>
    obtain ⟨U,hUx,hU⟩ := hX x
    obtain ⟨V,hVU,hV,hxV⟩ := mem_nhds_iff.mp hUx
    refine ⟨thickening a V,(thickening_isOpen a hV).mem_nhds hxV,
      thickening_disjoint a ha (fun g hg => hU g ?_)⟩
    exact hg.mono (Set.inter_subset_inter (Set.image_mono hVU) hVU)
  | inr p =>
    refine ⟨Set.range (openCell a p.1),(openCell_openEmbedding a p.1).isOpen_range.mem_nhds
      (Set.mem_range_self p.2),?_⟩
    intro g hg
    obtain ⟨_,⟨_,⟨x,rfl⟩,rfl⟩,⟨y,hy⟩⟩ := hg
    have hi : p.1 = g • p.1 := congrArg Sigma.fst (Sum.inr_injective hy)
    apply IsCancelSMul.right_cancel g 1 p.1
    simpa only [one_smul] using hi.symm

end GroupRingTopology.Attachment

end

end OAI

	import Mathlib
	import OAI.Algebra.GroupRing.HomotopyFunctor

	namespace OAI

	noncomputable section

	open Set Metric Topology BigOperators
	open Set Metric Topology

	namespace GroupRingTopology.Cellular
	universe u
	variable {X Y : Type u} [TopologicalSpace X] [TopologicalSpace Y] [Nonempty X]
	variable {C : Set X} [CWComplex C] {f : X → Y} (hf : IsClosedEmbedding f)

	def transportedMap (n : ℕ) (i : RelCWComplex.cell C n) : PartialEquiv (Fin n → ℝ) Y :=
	(RelCWComplex.map n i).trans (hf.isEmbedding.toPartialHomeomorph f).toPartialEquiv

	@[simp] lemma transportedMap_apply (n : ℕ) (i : RelCWComplex.cell C n) (x : Fin n → ℝ) :
	transportedMap hf n i x = f (RelCWComplex.map n i x) := rfl

	lemma transportedMap_source (n : ℕ) (i : RelCWComplex.cell C n) :
	(transportedMap hf n i).source = ball 0 1 := by
	simp only [transportedMap,PartialEquiv.trans_source,
	IsEmbedding.toPartialHomeomorph_source,Set.preimage_univ,Set.inter_univ,
	RelCWComplex.source_eq]

	lemma transportedMap_image (n : ℕ) (i : RelCWComplex.cell C n) (S : Set (Fin n → ℝ)) :
	transportedMap hf n i '' S = f '' (RelCWComplex.map n i '' S) :=
	Set.image_comp f (RelCWComplex.map n i) S

	lemma transportedMap_continuousOn (n : ℕ) (i : RelCWComplex.cell C n) :
	ContinuousOn (transportedMap hf n i) (closedBall 0 1) :=
	hf.continuous.comp_continuousOn (RelCWComplex.continuousOn n i)

	lemma transportedMap_continuousOn_symm (n : ℕ) (i : RelCWComplex.cell C n) :
	ContinuousOn (transportedMap hf n i).symm (transportedMap hf n i).target := by
	let e := hf.isEmbedding.toPartialHomeomorph f
	change ContinuousOn (fun y => (RelCWComplex.map n i).symm (e.symm y))
	(e.target ∩ e.symm ⁻¹' (RelCWComplex.map n i).target)
	exact (RelCWComplex.continuousOn_symm n i).comp
	(e.continuousOn_symm.mono Set.inter_subset_left) (fun _ h => h.2)

	variable [T2Space X]

	/-- Closed embeddings transport classical CW structures, including their weak
	topology, with precisely the same cells. -/
	@[instance_reducible]
	def transportCW : CWComplex (f '' C) where
	cell := RelCWComplex.cell C
	map := transportedMap hf
	source_eq := transportedMap_source hf
	continuousOn := transportedMap_continuousOn hf
	continuousOn_symm := transportedMap_continuousOn_symm hf
	pairwiseDisjoint' := by
	intro p _ q _ hpq
	change Disjoint (transportedMap hf p.1 p.2 '' ball 0 1)
	(transportedMap hf q.1 q.2 '' ball 0 1)
	rw [transportedMap_image,transportedMap_image]
	exact (CWComplex.disjoint_openCell_of_ne hpq).image hf.injective.injOn (Set.subset_univ _) (Set.subset_univ _)
	mapsTo' := by
	intro n i
	obtain ⟨I,hI⟩ := CWComplex.cellFrontier_subset_finite_closedCell n i
	refine ⟨I,?_⟩
	intro x hx
	have h := hI (show RelCWComplex.map n i x ∈ CWComplex.cellFrontier n i from ⟨x,hx,rfl⟩)
	simp only [Set.mem_iUnion] at h
	obtain ⟨m,hm,j,hj,hx⟩ := h
	refine Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨hm,Set.mem_iUnion.mpr
	⟨j,Set.mem_iUnion.mpr ⟨hj,?_⟩⟩⟩⟩
	rw [transportedMap_image]
	exact ⟨_,hx,rfl⟩
	closed' := by
	intro A hA hclosed
	have hpre : IsClosed (f ⁻¹' A) := by
	apply (CWComplex.closed C (f ⁻¹' A) ?_).mpr
	· intro n i
	have h := (hclosed n i).preimage hf.continuous
	rw [transportedMap_image,Set.preimage_inter,
	Set.preimage_image_eq _ hf.injective] at h
	exact h
	· intro x hx
	obtain ⟨y,hy,hyx⟩ := hA hx
	exact hf.injective hyx ▸ hy
	have h := hf.isClosedMap _ hpre
	rwa [Set.image_preimage_eq_of_subset (hA.trans (Set.image_subset_range f C))] at h
	union' := by
	simp only [transportedMap_image,← Set.image_iUnion]
	change f '' (⋃ n, ⋃ i, CWComplex.closedCell n i) = f '' C
	rw [CWComplex.union]

	end GroupRingTopology.Cellular
	open Set Metric Topology
	namespace GroupRingTopology.Attachment

	section
	open Set Metric Topology
	universe z
	variable {X : Type z} [TopologicalSpace X] [T2Space X] [Nonempty X]
	variable [CWComplex (Set.univ : Set X)] {ι : Type z} {n : ℕ}
	variable (a : ι → C(Sphere n,X))

	abbrev OldCell (m : ℕ) := RelCWComplex.cell (Set.univ : Set X) m

	omit [Nonempty X] in
	lemma attaching_finite_frontier
	(hdim : ∀ m (_ : OldCell (X := X) m), m < n) (i : ι) :
	∃ I : ∀ m, Finset (OldCell (X := X) m),
	∀ x, a i x ∈ ⋃ m < n, ⋃ j ∈ I m, CWComplex.closedCell m j := by
	classical
	let S := {p : Σ m, OldCell (X := X) m \|
	(Set.range (a i) ∩ CWComplex.openCell p.1 p.2).Nonempty}
	have hS : S.Finite := Cellular.compact_finitely_many_cells (C := (Set.univ : Set X))
	(isCompact_range (a i).continuous)
	have hSm (m : ℕ) : ((Sigma.mk m) ⁻¹' S).Finite :=
	hS.preimage (Set.injOn_of_injective (fun _ _ h => eq_of_heq (Sigma.mk.inj_iff.mp h).2))
	refine ⟨fun m => (hSm m).toFinset,?_⟩
	intro x
	have hx : a i x ∈ (Set.univ : Set X) := Set.mem_univ _
	rw [← CWComplex.iUnion_openCell_eq_complex (C := (Set.univ : Set X))] at hx
	simp only [Set.mem_iUnion] at hx
	obtain ⟨m,j,hx⟩ := hx
	exact Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨hdim m j,Set.mem_iUnion.mpr
	⟨j,Set.mem_iUnion.mpr ⟨(hSm m).mem_toFinset.mpr ⟨a i x,⟨x,rfl⟩,hx⟩,
	CWComplex.openCell_subset_closedCell m j hx⟩⟩⟩⟩

	abbrev LayerCell (m : ℕ) := OldCell (X := X) m ⊕ CellIndex.{z,z} m n ι

	def layerMap (m : ℕ) : LayerCell (X := X) (ι := ι) (n := n) m →
	PartialEquiv (Fin m → ℝ) (Space a)
	\| Sum.inl i => Cellular.transportedMap (C := Set.univ) (base_closedEmbedding a) m i
	\| Sum.inr i => cellMap a m i

	omit [T2Space X] in
	lemma layerMap_old_apply (m : ℕ) (i : OldCell (X := X) m) (x : Fin m → ℝ) :
	layerMap a m (Sum.inl i) x = base a (RelCWComplex.map m i x) := rfl

	omit [T2Space X] in
	lemma layerMap_old_image (m : ℕ) (i : OldCell (X := X) m) (S : Set (Fin m → ℝ)) :
	layerMap a m (Sum.inl i) '' S = base a '' (RelCWComplex.map m i '' S) :=
	Cellular.transportedMap_image (base_closedEmbedding a) m i S

	omit [T2Space X] in
	lemma layerMap_new (i : ι) :
	layerMap a n (Sum.inr ⟨⟨⟨rfl⟩,i⟩⟩) = characteristicPE a i := rfl

	@[instance_reducible]
	def layerCW (hdim : ∀ m (_ : OldCell (X := X) m), m < n) :
	CWComplex (Set.univ : Set (Space a)) where
	cell := LayerCell (X := X) (ι := ι) (n := n)
	map := layerMap a
	source_eq := by
	intro m i
	cases i with
	\| inl i => exact Cellular.transportedMap_source (base_closedEmbedding a) m i
	\| inr i => exact (relCWComplex a).source_eq m i
	continuousOn := by
	intro m i
	cases i with
	\| inl i => exact Cellular.transportedMap_continuousOn (base_closedEmbedding a) m i
	\| inr i => exact (relCWComplex a).continuousOn m i
	continuousOn_symm := by
	intro m i
	cases i with
	\| inl i => exact Cellular.transportedMap_continuousOn_symm (base_closedEmbedding a) m i
	\| inr i => exact (relCWComplex a).continuousOn_symm m i
	pairwiseDisjoint' := by
	rintro ⟨m,i⟩ _ ⟨k,j⟩ _ hne
	change Disjoint (layerMap a m i '' ball 0 1) (layerMap a k j '' ball 0 1)
	cases i with
	\| inl i =>
	cases j with
	\| inl j =>
	rw [layerMap_old_image,layerMap_old_image]
	apply (CWComplex.disjoint_openCell_of_ne ?_).image (base_closedEmbedding a).injective.injOn (Set.subset_univ _) (Set.subset_univ _)
	intro h
	cases h
	exact hne rfl
	\| inr j =>
	obtain ⟨⟨⟨hk⟩,j⟩⟩ := j
	subst k
	rw [layerMap_old_image,layerMap_new,characteristicPE_image_ball]
	apply Set.disjoint_left.mpr
	rintro z ⟨x,_,rfl⟩ ⟨y,h⟩
	cases h
	\| inr i =>
	obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
	subst m
	cases j with
	\| inl j =>
	rw [layerMap_new,characteristicPE_image_ball,layerMap_old_image]
	apply Set.disjoint_left.mpr
	rintro z ⟨x,rfl⟩ ⟨y,_,h⟩
	cases h
	\| inr j =>
	obtain ⟨⟨⟨hk⟩,j⟩⟩ := j
	subst k
	rw [layerMap_new,layerMap_new,characteristicPE_image_ball,characteristicPE_image_ball]
	apply Set.disjoint_left.mpr
	rintro z ⟨x,rfl⟩ ⟨y,h⟩
	have hij : j = i := congrArg Sigma.fst (Sum.inr_injective h)
	subst j
	exact hne rfl
	mapsTo' := by
	classical
	intro m c
	cases c with
	\| inl i =>
	obtain ⟨I,hI⟩ := CWComplex.cellFrontier_subset_finite_closedCell m i
	refine ⟨fun k => (I k).image Sum.inl,?_⟩
	intro x hx
	have hh := hI (show RelCWComplex.map m i x ∈ CWComplex.cellFrontier m i from ⟨x,hx,rfl⟩)
	simp only [Set.mem_iUnion] at hh
	obtain ⟨k,hkm,j,hj,hx⟩ := hh
	refine Set.mem_iUnion.mpr ⟨k,Set.mem_iUnion.mpr ⟨hkm,Set.mem_iUnion.mpr
	⟨Sum.inl j,Set.mem_iUnion.mpr ⟨Finset.mem_image.mpr ⟨j,hj,rfl⟩,?_⟩⟩⟩⟩
	rw [layerMap_old_image]
	exact ⟨_,hx,rfl⟩
	\| inr i =>
	obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
	subst m
	obtain ⟨I,hI⟩ := attaching_finite_frontier a hdim i
	refine ⟨fun k => (I k).image Sum.inl,?_⟩
	intro x hx
	have hh := hI ⟨x,hx⟩
	simp only [Set.mem_iUnion] at hh
	obtain ⟨k,hkn,j,hj,hx'⟩ := hh
	refine Set.mem_iUnion.mpr ⟨k,Set.mem_iUnion.mpr ⟨hkn,Set.mem_iUnion.mpr
	⟨Sum.inl j,Set.mem_iUnion.mpr ⟨Finset.mem_image.mpr ⟨j,hj,rfl⟩,?_⟩⟩⟩⟩
	rw [layerMap_old_image]
	refine ⟨a i ⟨x,hx⟩,hx',?_⟩
	change base a (a i ⟨x,hx⟩) = characteristic a i x
	exact (cell_boundary a i ⟨x,hx⟩).symm.trans
	(characteristic_disk a i (boundary n ⟨x,hx⟩)).symm
	closed' := by
	intro A _ hA
	apply (isClosed_iff a A).mpr
	constructor
	· apply (CWComplex.closed Set.univ (base a ⁻¹' A) (Set.subset_univ _)).mpr
	intro m i
	have h := (hA m (Sum.inl i)).preimage (continuous_base a)
	rw [layerMap_old_image,Set.preimage_inter,
	Set.preimage_image_eq _ (base_closedEmbedding a).injective] at h
	exact h
	· intro i
	have h := (hA n (Sum.inr ⟨⟨⟨rfl⟩,i⟩⟩)).preimage (cell a i).continuous
	rw [layerMap_new,characteristicPE_image_closedBall,Set.preimage_inter] at h
	simpa only [Set.preimage_range,Set.inter_univ] using h
	union' := by
	apply Set.eq_univ_of_forall
	intro z
	cases z with
	\| inl x =>
	have hx : x ∈ (Set.univ : Set X) := Set.mem_univ _
	rw [← CWComplex.union (C := (Set.univ : Set X))] at hx
	simp only [Set.mem_iUnion] at hx
	obtain ⟨m,i,hx⟩ := hx
	refine Set.mem_iUnion.mpr ⟨m,Set.mem_iUnion.mpr ⟨Sum.inl i,?_⟩⟩
	rw [layerMap_old_image]
	exact ⟨x,hx,rfl⟩
	\| inr p =>
	refine Set.mem_iUnion.mpr ⟨n,Set.mem_iUnion.mpr ⟨Sum.inr ⟨⟨⟨rfl⟩,p.1⟩⟩,?_⟩⟩
	rw [layerMap_new,characteristicPE_image_closedBall]
	exact ⟨interior n p.2,cell_interior a p.1 p.2⟩

	lemma layerCW_dimensions (hdim : ∀ m (_ : OldCell (X := X) m), m < n)
	(m : ℕ) (i : (layerCW a hdim).cell m) : m < n+1 := by
	cases i with
	\| inl i => exact (hdim m i).trans (Nat.lt_succ_self n)
	\| inr i =>
	obtain ⟨⟨⟨hm⟩,i⟩⟩ := i
	subst m
	exact Nat.lt_succ_self n

	lemma layerCW_finite_cells [Finite ι]
	(hfin : ∀ m, Finite (OldCell (X := X) m))
	(hdim : ∀ m (_ : OldCell (X := X) m), m < n) (m : ℕ) :
	Finite ((layerCW a hdim).cell m) := by
	let := hfin m
	change Finite (LayerCell (X := X) (ι := ι) (n := n) m)
	infer_instance

	end

	section
	open Set Metric Topology
	universe u v w z
	variable {X : Type u} [TopologicalSpace X] {ι : Type v} {n : ℕ}
	variable (a : ι → C(Sphere n,X))
	variable {G : Type w} [Group G] [MulAction G X] [MulAction G ι]

	instance spaceMulAction : MulAction G (Space a) where
	smul g := Sum.elim (fun x => Sum.inl (g • x)) (fun p => Sum.inr ⟨g • p.1,p.2⟩)
	one_smul := by
	intro x
	cases x with
	\| inl x => change Sum.inl ((1 : G) • x) = Sum.inl x; rw [one_smul]
	\| inr p => change Sum.inr ⟨(1 : G) • p.1,p.2⟩ = Sum.inr p; rw [one_smul]
	mul_smul := by
	intro g h x
	cases x with
	\| inl x => change Sum.inl ((g * h) • x) = Sum.inl (g • h • x); rw [mul_smul]
	\| inr p => change (Sum.inr ⟨(g * h) • p.1,p.2⟩ : Space a) = Sum.inr ⟨g • h • p.1,p.2⟩; rw [mul_smul]

	@[simp] lemma smul_base (g : G) (x : X) : g • base a x = base a (g • x) := rfl
	@[simp] lemma smul_openCell (g : G) (i : ι) (x : OpenDisk n) :
	g • openCell a i x = openCell a (g • i) x := rfl

	variable (ha : ∀ (g : G) i x, a (g • i) x = g • a i x)
	include ha
	lemma smul_cell (g : G) (i : ι) (d : Disk n) :
	g • cell a i d = cell a (g • i) d := by
	classical
	by_cases hd : d.1 ∈ ball (0 : Fin n → ℝ) 1
	· have heq : d = interior n ⟨d.1,hd⟩ := rfl
	rw [heq,cell_interior,cell_interior]
	rfl
	· let x : Sphere n := ⟨d.1,by
	have hm := d.2
	simp only [mem_closedBall,mem_ball,mem_sphere] at *
	exact le_antisymm hm (le_of_not_gt hd)⟩
	have heq : d = boundary n x := rfl
	rw [heq,cell_boundary,cell_boundary,smul_base,ha]

	lemma continuousConstSMul [ContinuousConstSMul G X] : ContinuousConstSMul G (Space a) where
	continuous_const_smul g := (continuous_iff a _).mpr ⟨
	(continuous_base a).comp (continuous_const_smul g),fun i => by
	have heq : (g • ·) ∘ cell a i = cell a (g • i) := funext (smul_cell a ha g i)
	rw [heq]
	exact (cell a (g • i)).continuous⟩

	omit ha in
	lemma isOpen_iff (S : Set (Space a)) :
	IsOpen S ↔ IsOpen (base a ⁻¹' S) ∧ ∀ i, IsOpen (cell a i ⁻¹' S) := by
	change @IsOpen _ (TopologicalSpace.coinduced (quotient a) inferInstance) S ↔ _
	rw [isOpen_coinduced,isOpen_sum_iff,isOpen_sigma_iff]
	rfl

	end

	open Set Metric Topology
	universe u v w
	abbrev Collar (n : ℕ) := {d : Disk n \| (1 / 2 : ℝ) < ‖d.1‖}

	def collarInclusion (n : ℕ) : Collar n → Disk n := Subtype.val

	def radial (n : ℕ) : Collar n → Sphere n := fun d =>
	⟨‖d.1.1‖⁻¹ • d.1.1,by
	have hn : ‖d.1.1‖ ≠ 0 := ne_of_gt (lt_trans (by norm_num) d.2)
	rw [mem_sphere,dist_zero_right,norm_smul,Real.norm_eq_abs,abs_inv,abs_norm,
	inv_mul_cancel₀ hn]⟩

	lemma continuous_radial (n : ℕ) : Continuous (radial n) := by
	apply Continuous.subtype_mk
	have hn : ∀ d : Collar n, ‖d.1.1‖ ≠ 0 := fun d => ne_of_gt (lt_trans (by norm_num) d.2)
	exact ((continuous_subtype_val.comp continuous_subtype_val).norm.inv₀ hn).smul
	(continuous_subtype_val.comp continuous_subtype_val)

	lemma collar_openEmbedding (n : ℕ) : IsOpenEmbedding (collarInclusion n) :=
	(isOpen_lt continuous_const continuous_subtype_val.norm).isOpenEmbedding_subtypeVal

	lemma radial_boundary (n : ℕ) (x : Sphere n) (h : (1/2:ℝ) < ‖(boundary n x).1‖) :
	radial n ⟨boundary n x,h⟩ = x := by
	apply Subtype.ext
	have hx : ‖x.1‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using x.2
	simp only [radial,boundary,hx,inv_one,one_smul]

	variable {X : Type u} [TopologicalSpace X] {ι : Type v} {n : ℕ}
	variable (a : ι → C(Sphere n,X))

	/-- An open old-space neighborhood thickened by the outer half of each disk.
	Membership is tested by radial retraction of the disk to its attaching sphere. -/
	def thickening (U : Set X) : Set (Space a) := fun z =>
	match z with
	\| Sum.inl x => x ∈ U
	\| Sum.inr p => ∃ h : (1/2:ℝ) < ‖p.2.1‖,
	a p.1 (radial n ⟨interior n p.2,h⟩) ∈ U

	@[simp] lemma base_mem_thickening (U : Set X) (x : X) :
	base a x ∈ thickening a U ↔ x ∈ U := Iff.rfl

	lemma cell_preimage_thickening (U : Set X) (i : ι) :
	cell a i ⁻¹' thickening a U =
	collarInclusion n '' ((a i ∘ radial n) ⁻¹' U) := by
	classical
	ext d
	by_cases hd : d.1 ∈ ball (0 : Fin n → ℝ) 1
	· have heq : d = interior n ⟨d.1,hd⟩ := rfl
	rw [heq]
	simp only [Set.mem_preimage,cell_interior,Set.mem_image]
	constructor
	· rintro ⟨h,hh⟩
	exact ⟨⟨d,h⟩,hh,rfl⟩
	· rintro ⟨y,hy,he⟩
	have heq' : y.1 = d := he
	subst d
	exact ⟨y.2,hy⟩
	· let x : Sphere n := ⟨d.1,by
	have hm := d.2
	simp only [mem_closedBall,mem_ball,mem_sphere] at *
	exact le_antisymm hm (le_of_not_gt hd)⟩
	have heq : d = boundary n x := rfl
	have hx : ‖x.1‖ = 1 := by simpa only [mem_sphere,dist_zero_right] using x.2
	have hhalf : (1/2:ℝ) < ‖(boundary n x).1‖ := by change (1/2:ℝ) < ‖x.1‖; rw [hx]; norm_num
	rw [heq]
	simp only [Set.mem_preimage,cell_boundary,base_mem_thickening,Set.mem_image]
	constructor
	· intro h
	exact ⟨⟨boundary n x,hhalf⟩,by simpa only [Function.comp_apply,radial_boundary] using h,rfl⟩
	· rintro ⟨y,hy,he⟩
	have heq' : y = ⟨boundary n x,hhalf⟩ := Subtype.ext he
	simpa only [heq',Function.comp_apply,radial_boundary] using hy

	lemma thickening_isOpen {U : Set X} (hU : IsOpen U) : IsOpen (thickening a U) := by
	rw [isOpen_iff]
	refine ⟨hU,fun i => ?_⟩
	rw [cell_preimage_thickening]
	exact (collar_openEmbedding n).isOpenMap _
	(hU.preimage ((a i).continuous.comp (continuous_radial n)))

	variable {G : Type w} [Group G] [MulAction G X] [MulAction G ι]
	variable (ha : ∀ (g : G) i x, a (g • i) x = g • a i x)
	include ha
	lemma thickening_disjoint {U : Set X}
	(hU : ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1) :
	∀ g : G, ((g • ·) '' thickening a U ∩ thickening a U).Nonempty → g = 1 := by
	intro g h
	obtain ⟨_,⟨z,hz,rfl⟩,hgz⟩ := h
	cases z with
	\| inl x => exact hU g ⟨g • x,⟨x,hz,rfl⟩,hgz⟩
	\| inr p =>
	obtain ⟨hhalf,hz⟩ := hz
	obtain ⟨_,hgz⟩ := hgz
	change a (g • p.1) (radial n ⟨interior n p.2,hhalf⟩) ∈ U at hgz
	rw [ha] at hgz
	exact hU g ⟨g • a p.1 (radial n ⟨interior n p.2,hhalf⟩),⟨_,hz,rfl⟩,hgz⟩

	lemma exists_disjoint_nhds [IsCancelSMul G ι]
	(hX : ∀ x : X, ∃ U ∈ 𝓝 x, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1) :
	∀ z : Space a, ∃ U ∈ 𝓝 z, ∀ g : G, ((g • ·) '' U ∩ U).Nonempty → g = 1 := by
	intro z
	cases z with
	\| inl x =>
	obtain ⟨U,hUx,hU⟩ := hX x
	obtain ⟨V,hVU,hV,hxV⟩ := mem_nhds_iff.mp hUx
	refine ⟨thickening a V,(thickening_isOpen a hV).mem_nhds hxV,
	thickening_disjoint a ha (fun g hg => hU g ?_)⟩
	exact hg.mono (Set.inter_subset_inter (Set.image_mono hVU) hVU)
	\| inr p =>
	refine ⟨Set.range (openCell a p.1),(openCell_openEmbedding a p.1).isOpen_range.mem_nhds
	(Set.mem_range_self p.2),?_⟩
	intro g hg
	obtain ⟨_,⟨_,⟨x,rfl⟩,rfl⟩,⟨y,hy⟩⟩ := hg
	have hi : p.1 = g • p.1 := congrArg Sigma.fst (Sum.inr_injective hy)
	apply IsCancelSMul.right_cancel g 1 p.1
	simpa only [one_smul] using hi.symm

	end GroupRingTopology.Attachment

	end

	end OAI