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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 | import Mathlib
import OAI.Geometry.HyperbolicGroups.HSpacePathObstruction
import OAI.Geometry.HyperbolicGroups.PathHomotopyTopology
import OAI.Geometry.HyperbolicGroups.ParameterizedSquareFilling
namespace OAI
/-! A vanishing actual pi-2 obstruction constructs a continuous square
contraction with four independently prescribed edge contractions. -/
noncomputable section
open scoped Topology unitInterval
namespace HyperbolicObstruction.BasedSquareContraction
open HSpacePathObstruction
variable {X : Type*} [TopologicalSpace X] (b : X)
variable {p q r s : Path b b}
variable (f : C(I × I,X))
(hb : ∀ x, f (x,0) = p x) (hl : ∀ x, f (0,x) = q x)
(hr : ∀ x, f (1,x) = r x) (ht : ∀ x, f (x,1) = s x)
def facePath : Path (p.trans r) (q.trans s) :=
PathHomotopyTopology.fromHomotopy (SquarePathFilling.unfill f hb hl hr ht)
variable (hp : Path p (Path.refl b)) (hq : Path q (Path.refl b)) (hr' : Path r (Path.refl b)) (hs : Path s (Path.refl b))
def faceObstruction : FundamentalGroup (Path b b) (Path.refl b) :=
obstruction (facePath b f hb hl hr ht) hp hr' hq hs
variable (hz : faceObstruction b f hb hl hr ht hp hq hr' hs = 1)
def loopSquare : C(I × I,Path b b) :=
SquarePathFilling.fill
((obstruction_vanishes_iff (facePath b f hb hl hr ht) hp hr' hq hs).mp hz).some
@[simp] lemma loopSquare_bottom (t : I) :
loopSquare b f hb hl hr ht hp hq hr' hs hz (t,0) = (hp t).trans (hr' t) :=
SquarePathFilling.fill_bottom _ t
@[simp] lemma loopSquare_top (t : I) :
loopSquare b f hb hl hr ht hp hq hr' hs hz (t,1) = (hq t).trans (hs t) :=
SquarePathFilling.fill_top _ t
@[simp] lemma loopSquare_left (t : I) :
loopSquare b f hb hl hr ht hp hq hr' hs hz (0,t) = facePath b f hb hl hr ht t :=
SquarePathFilling.fill_left _ t
@[simp] lemma loopSquare_right (t : I) :
loopSquare b f hb hl hr ht hp hq hr' hs hz (1,t) = Path.refl b :=
SquarePathFilling.fill_right _ t
/-- Literal face homotopy. Its domain is the actual closed square prism. -/
def contraction : C(I × (I × I),X) :=
SquarePathFilling.paramFill b (loopSquare b f hb hl hr ht hp hq hr' hs hz)
@[simp] lemma contraction_initial (z : I × I) :
contraction b f hb hl hr ht hp hq hr' hs hz (0,z) = f z := by
obtain ⟨z,rfl⟩ := SquarePathFilling.quotient_surjective z
unfold contraction
rw [SquarePathFilling.paramFill_quotient,loopSquare_left]
rfl
@[simp] lemma contraction_final (z : I × I) :
contraction b f hb hl hr ht hp hq hr' hs hz (1,z) = b := by
obtain ⟨z,rfl⟩ := SquarePathFilling.quotient_surjective z
unfold contraction
rw [SquarePathFilling.paramFill_quotient,loopSquare_right]
rfl
@[simp] lemma contraction_bottom (t x : I) :
contraction b f hb hl hr ht hp hq hr' hs hz (t,(x,0)) = hp t x := by
unfold contraction
rw [←SquarePathFilling.quotient_bottom x,SquarePathFilling.paramFill_quotient,loopSquare_bottom]
exact SquarePathFilling.trans_half _ _ x
@[simp] lemma contraction_right (t x : I) :
contraction b f hb hl hr ht hp hq hr' hs hz (t,(1,x)) = hr' t x := by
unfold contraction
rw [←SquarePathFilling.quotient_right x,SquarePathFilling.paramFill_quotient,loopSquare_bottom]
exact SquarePathFilling.trans_upper _ _ x
@[simp] lemma contraction_left (t x : I) :
contraction b f hb hl hr ht hp hq hr' hs hz (t,(0,x)) = hq t x := by
unfold contraction
rw [←SquarePathFilling.quotient_left x,SquarePathFilling.paramFill_quotient,loopSquare_top]
exact SquarePathFilling.trans_half _ _ x
@[simp] lemma contraction_top (t x : I) :
contraction b f hb hl hr ht hp hq hr' hs hz (t,(x,1)) = hs t x := by
unfold contraction
rw [←SquarePathFilling.quotient_top x,SquarePathFilling.paramFill_quotient,loopSquare_top]
exact SquarePathFilling.trans_upper _ _ x
end HyperbolicObstruction.BasedSquareContraction
end
end OAI
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