File size: 4,283 Bytes
7d9acbd | 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 | import Mathlib
import OAI.Geometry.HyperbolicGroups.TriangleSquare
import OAI.Geometry.HyperbolicGroups.BasedSquareContraction
namespace OAI
/-! The actual triangular pi2 obstruction has a literal contracting prism
when the three oriented edge corrections have the prescribed coboundary. -/
noncomputable section
open scoped Topology unitInterval
namespace HyperbolicObstruction.BasedTriangleContraction
open TriangleSquare HSpacePathObstruction
variable {X : Type*} [TopologicalSpace X] (b : X)
local notation "e" => Path.refl b
variable (f : C(Triangle,X))
(hb : ∀ s, f (quotient (s,0)) = b)
(hl : ∀ s, f (quotient (0,s)) = b)
(hr : ∀ s, f (quotient (1,s)) = b)
include hr in
lemma top_const (s : I) : f (quotient (s,1)) = b := by
rw [quotient_one,←quotient_one (1 : I)]
exact hr 1
def theta : FundamentalGroup (Path b b) e :=
BasedSquareContraction.faceObstruction b (p:=e) (q:=e) (r:=e) (s:=e) (f.comp quotient) hb hl hr (top_const b f hr)
(Path.refl e) (Path.refl e) (Path.refl e) (Path.refl e)
variable (u01 u02 u12 : Path (Path.refl b) (Path.refl b))
(hu : loopClass u02 * theta b f hb hl hr = loopClass u01 * loopClass u12)
lemma corrected_zero (hu : loopClass u02 * theta b f hb hl hr = loopClass u01 * loopClass u12) :
BasedSquareContraction.faceObstruction b (p:=e) (q:=e) (r:=e) (s:=e) (f.comp quotient) hb hl hr (top_const b f hr)
((Path.refl e).trans u01) ((Path.refl e).trans u02)
((Path.refl e).trans u12) ((Path.refl e).trans (Path.refl e)) = 1 := by
unfold BasedSquareContraction.faceObstruction
rw [correction]
have he : loopClass (Path.refl e) = 1 := rfl
rw [he,mul_one]
change (loopClass u02 * theta b f hb hl hr) * (loopClass u01 * loopClass u12)⁻¹ = 1
rw [hu,mul_inv_cancel]
def square : C(I × (I × I),X) :=
BasedSquareContraction.contraction b (p:=e) (q:=e) (r:=e) (s:=e) (f.comp quotient) hb hl hr (top_const b f hr)
((Path.refl e).trans u01) ((Path.refl e).trans u02)
((Path.refl e).trans u12) ((Path.refl e).trans (Path.refl e))
(corrected_zero b f hb hl hr u01 u02 u12 hu)
lemma square_top (t s : I) : square b f hb hl hr u01 u02 u12 hu (t,(s,1)) = b := by
calc
_ = ((Path.refl e).trans (Path.refl e)) t s :=
BasedSquareContraction.contraction_top b (p := e) (q := e) (r := e) (s := e)
(f.comp quotient) hb hl hr (top_const b f hr)
((Path.refl e).trans u01) ((Path.refl e).trans u02)
((Path.refl e).trans u12) ((Path.refl e).trans (Path.refl e))
(corrected_zero b f hb hl hr u01 u02 u12 hu) t s
_ = b := congrArg (fun p : Path e e => p t s) Path.refl_trans_refl
def contraction : C(I × Triangle,X) := TriangleSquare.descend
(square b f hb hl hr u01 u02 u12 hu)
(fun t s => (square_top b f hb hl hr u01 u02 u12 hu t s).trans
(square_top b f hb hl hr u01 u02 u12 hu t 0).symm)
lemma contraction_initial (z : Triangle) : contraction b f hb hl hr u01 u02 u12 hu (0,z) = f z := by
obtain ⟨z,rfl⟩ := quotient_surjective z
unfold contraction
rw [TriangleSquare.descend_quotient]
exact BasedSquareContraction.contraction_initial _ _ _ _ _ _ _ _ _ _ _ z
lemma contraction_final (z : Triangle) : contraction b f hb hl hr u01 u02 u12 hu (1,z) = b := by
obtain ⟨z,rfl⟩ := quotient_surjective z
unfold contraction
rw [TriangleSquare.descend_quotient]
exact BasedSquareContraction.contraction_final _ _ _ _ _ _ _ _ _ _ _ z
lemma contraction_bottom (t s : I) :
contraction b f hb hl hr u01 u02 u12 hu (t,quotient (s,0)) = ((Path.refl e).trans u01) t s := by
unfold contraction
rw [TriangleSquare.descend_quotient]
exact BasedSquareContraction.contraction_bottom _ _ _ _ _ _ _ _ _ _ _ t s
lemma contraction_left (t s : I) :
contraction b f hb hl hr u01 u02 u12 hu (t,quotient (0,s)) = ((Path.refl e).trans u02) t s := by
unfold contraction
rw [TriangleSquare.descend_quotient]
exact BasedSquareContraction.contraction_left _ _ _ _ _ _ _ _ _ _ _ t s
lemma contraction_right (t s : I) :
contraction b f hb hl hr u01 u02 u12 hu (t,quotient (1,s)) = ((Path.refl e).trans u12) t s := by
unfold contraction
rw [TriangleSquare.descend_quotient]
exact BasedSquareContraction.contraction_right _ _ _ _ _ _ _ _ _ _ _ t s
end HyperbolicObstruction.BasedTriangleContraction
end
end OAI
|