990 lines
40 KiB
Text
990 lines
40 KiB
Text
/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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We define the fiber sequence of a pointed map f : X →* Y. We follow the proof in section 8.4 of
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the book closely. First we define a sequence fiber_sequence as in Definition 8.4.3.
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It has types X(n) : Type*
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X(0) := Y,
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X(1) := X,
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X(n+1) := pfiber (f(n))
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with functions f(n) : X(n+1) →* X(n)
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f(0) := f
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f(n+1) := ppoint f(n)
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We prove that this is an exact sequence.
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Then we prove Lemma 8.4.3, by showing that X(n+3) ≃* Ω(X(n)) and that this equivalence sends
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the map f(n+3) to -Ω(f(n)), i.e. the composition of Ω(f(n)) with path inversion.
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This is the hardest part of this formalization, because we need to show that they are the same
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as pointed maps (we define a pointed homotopy between them).
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Using this equivalence we get a boundary_map : Ω(Y) → pfiber f and we can define a new
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fiber sequence X'(n) : Type*
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X'(0) := Y,
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X'(1) := X,
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X'(2) := pfiber f
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X'(n+3) := Ω(X'(n))
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and maps f'(n) : X'(n+1) →* X'(n)
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f'(0) := f
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f'(1) := ppoint f
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f'(2) := boundary_map f
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f'(3) := -Ω(f)
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f'(4) := -Ω(ppoint f)
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f'(5) := -Ω(boundary_map f)
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f'(n+6) := Ω²(f'(n))
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We can show that these sequences are equivalent, hence the sequence (X',f') is an exact
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sequence. Now we get the fiber sequence by taking the set-truncation of this sequence.
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-/
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import .chain_complex algebra.homotopy_group
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open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function
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/--------------
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PART 1
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--------------/
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namespace chain_complex
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definition fiber_sequence_helper [constructor] (v : Σ(X Y : Type*), X →* Y)
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: Σ(Z X : Type*), Z →* X :=
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⟨pfiber v.2.2, v.1, ppoint v.2.2⟩
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definition fiber_sequence_helpern (v : Σ(X Y : Type*), X →* Y) (n : ℕ)
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: Σ(Z X : Type*), Z →* X :=
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iterate fiber_sequence_helper n v
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universe variable u
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variables {X Y : pType.{u}} (f : X →* Y) (n : ℕ)
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include f
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definition fiber_sequence_carrier : Type* :=
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(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.1
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definition fiber_sequence_fun
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: fiber_sequence_carrier f (n + 1) →* fiber_sequence_carrier f n :=
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(fiber_sequence_helpern ⟨X, Y, f⟩ n).2.2
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/- Definition 8.4.3 -/
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definition fiber_sequence : type_chain_complex.{0 u} +ℕ :=
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begin
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fconstructor,
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{ exact fiber_sequence_carrier f},
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{ exact fiber_sequence_fun f},
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{ intro n x, cases n with n,
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{ exact point_eq x},
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{ exact point_eq x}}
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end
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definition is_exact_fiber_sequence : is_exact_t (fiber_sequence f) :=
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λn x p, fiber.mk (fiber.mk x p) rfl
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/- (generalization of) Lemma 8.4.4(i)(ii) -/
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definition fiber_sequence_carrier_equiv
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: fiber_sequence_carrier f (n+3) ≃ Ω(fiber_sequence_carrier f n) :=
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calc
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fiber_sequence_carrier f (n+3) ≃ fiber (fiber_sequence_fun f (n+1)) pt : erfl
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... ≃ Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f (n+1) x = pt
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: fiber.sigma_char
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... ≃ Σ(x : fiber (fiber_sequence_fun f n) pt), fiber_sequence_fun f _ x = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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fiber_sequence_fun f _ (fiber.mk v.1 v.2) = pt
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: by exact sigma_equiv_sigma !fiber.sigma_char (λa, erfl)
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), fiber_sequence_fun f _ x = pt),
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v.1 = pt
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: erfl
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... ≃ Σ(v : Σ(x : fiber_sequence_carrier f _), x = pt),
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fiber_sequence_fun f _ v.1 = pt
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: sigma_assoc_comm_equiv
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... ≃ fiber_sequence_fun f _ !center.1 = pt
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: @(sigma_equiv_of_is_contr_left _) !is_contr_sigma_eq'
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... ≃ fiber_sequence_fun f _ pt = pt
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: erfl
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... ≃ pt = pt
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: by exact !equiv_eq_closed_left !respect_pt
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... ≃ Ω(fiber_sequence_carrier f n) : erfl
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/- computation rule -/
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definition fiber_sequence_carrier_equiv_eq
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(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
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(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
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: fiber_sequence_carrier_equiv f n (fiber.mk (fiber.mk x p) q)
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= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
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begin
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refine _ ⬝ !con.assoc⁻¹,
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apply whisker_left,
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refine transport_eq_Fl _ _ ⬝ _,
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apply whisker_right,
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refine inverse2 !ap_inv ⬝ !inv_inv ⬝ _,
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refine ap_compose (fiber_sequence_fun f n) pr₁ _ ⬝
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ap02 (fiber_sequence_fun f n) !ap_pr1_center_eq_sigma_eq',
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end
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definition fiber_sequence_carrier_equiv_inv_eq
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(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_equiv f n)⁻¹ᵉ p =
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fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
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begin
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apply inv_eq_of_eq,
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refine _ ⬝ !fiber_sequence_carrier_equiv_eq⁻¹, esimp,
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exact !inv_con_cancel_left⁻¹
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end
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definition fiber_sequence_carrier_pequiv
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: fiber_sequence_carrier f (n+3) ≃* Ω(fiber_sequence_carrier f n) :=
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pequiv_of_equiv (fiber_sequence_carrier_equiv f n)
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begin
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esimp,
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apply con.left_inv
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end
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definition fiber_sequence_carrier_pequiv_eq
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(x : fiber_sequence_carrier f (n+1)) (p : fiber_sequence_fun f n x = pt)
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(q : fiber_sequence_fun f (n+1) (fiber.mk x p) = pt)
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: fiber_sequence_carrier_pequiv f n (fiber.mk (fiber.mk x p) q)
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= !respect_pt⁻¹ ⬝ ap (fiber_sequence_fun f n) q⁻¹ ⬝ p :=
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fiber_sequence_carrier_equiv_eq f n x p q
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definition fiber_sequence_carrier_pequiv_inv_eq
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(p : Ω(fiber_sequence_carrier f n)) : (fiber_sequence_carrier_pequiv f n)⁻¹ᵉ* p =
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fiber.mk (fiber.mk pt (respect_pt (fiber_sequence_fun f n) ⬝ p)) idp :=
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fiber_sequence_carrier_equiv_inv_eq f n p
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attribute pequiv._trans_of_to_pmap [unfold 3]
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/- Lemma 8.4.4(iii) -/
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definition fiber_sequence_fun_eq_helper
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(p : Ω(fiber_sequence_carrier f (n + 1))) :
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fiber_sequence_carrier_pequiv f n
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(fiber_sequence_fun f (n + 3)
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((fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* p)) =
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ap1 (fiber_sequence_fun f n) p⁻¹ :=
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begin
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refine ap (λx, fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x))
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(fiber_sequence_carrier_pequiv_inv_eq f (n+1) p) ⬝ _,
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/- the following three lines are rewriting some reflexivities: -/
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-- replace (n + 3) with (n + 2 + 1),
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-- refine ap (fiber_sequence_carrier_pequiv f n)
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-- (fiber_sequence_fun_eq1 f (n+2) _ idp) ⬝ _,
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refine fiber_sequence_carrier_pequiv_eq f n pt (respect_pt (fiber_sequence_fun f n)) _ ⬝ _,
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esimp,
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apply whisker_right,
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apply whisker_left,
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apply ap02, apply inverse2, apply idp_con,
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end
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theorem fiber_sequence_carrier_pequiv_eq_point_eq_idp :
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fiber_sequence_carrier_pequiv_eq f n
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(Point (fiber_sequence_carrier f (n+1)))
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(respect_pt (fiber_sequence_fun f n))
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(respect_pt (fiber_sequence_fun f (n + 1))) = idp :=
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begin
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apply con_inv_eq_idp,
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refine ap (λx, whisker_left _ (_ ⬝ x)) _ ⬝ _,
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{ reflexivity},
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{ reflexivity},
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esimp,
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refine ap (whisker_left _)
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(transport_eq_Fl_idp_left (fiber_sequence_fun f n)
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(respect_pt (fiber_sequence_fun f n))) ⬝ _,
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apply whisker_left_idp_con_eq_assoc
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end
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theorem fiber_sequence_fun_phomotopy_helper :
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(fiber_sequence_carrier_pequiv f n ∘*
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fiber_sequence_fun f (n + 3)) ∘*
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(fiber_sequence_carrier_pequiv f (n + 1))⁻¹ᵉ* ~*
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ap1 (fiber_sequence_fun f n) ∘* pinverse :=
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begin
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fapply phomotopy.mk,
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{ exact (fiber_sequence_fun_eq_helper f n)},
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{ esimp, rewrite [idp_con], refine _ ⬝ whisker_left _ !idp_con⁻¹,
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apply whisker_right,
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apply whisker_left,
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exact fiber_sequence_carrier_pequiv_eq_point_eq_idp f n}
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end
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theorem fiber_sequence_fun_eq : Π(x : fiber_sequence_carrier f (n + 4)),
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fiber_sequence_carrier_pequiv f n (fiber_sequence_fun f (n + 3) x) =
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ap1 (fiber_sequence_fun f n) (fiber_sequence_carrier_pequiv f (n + 1) x)⁻¹ :=
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homotopy_of_inv_homotopy
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(pequiv.to_equiv (fiber_sequence_carrier_pequiv f (n + 1)))
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(fiber_sequence_fun_eq_helper f n)
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theorem fiber_sequence_fun_phomotopy :
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fiber_sequence_carrier_pequiv f n ∘*
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fiber_sequence_fun f (n + 3) ~*
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(ap1 (fiber_sequence_fun f n) ∘* pinverse) ∘* fiber_sequence_carrier_pequiv f (n + 1) :=
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begin
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apply phomotopy_of_pinv_right_phomotopy,
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apply fiber_sequence_fun_phomotopy_helper
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end
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definition boundary_map : Ω Y →* pfiber f :=
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fiber_sequence_fun f 2 ∘* (fiber_sequence_carrier_pequiv f 0)⁻¹ᵉ*
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/--------------
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PART 2
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--------------/
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/- Now we are ready to define the long exact sequence of homotopy groups.
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First we define its carrier -/
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definition homotopy_groups : ℕ → Type*
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| 0 := Y
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| 1 := X
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| 2 := pfiber f
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| (k+3) := Ω (homotopy_groups k)
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definition homotopy_groups_add3 [unfold_full] :
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homotopy_groups f (n+3) = Ω (homotopy_groups f n) :=
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by reflexivity
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definition homotopy_groups_mul3
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: Πn, homotopy_groups f (3 * n) = Ω[n] Y :> Type*
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| 0 := proof rfl qed
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3 k) qed
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definition homotopy_groups_mul3add1
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: Πn, homotopy_groups f (3 * n + 1) = Ω[n] X :> Type*
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| 0 := by reflexivity
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3add1 k) qed
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definition homotopy_groups_mul3add2
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: Πn, homotopy_groups f (3 * n + 2) = Ω[n] (pfiber f) :> Type*
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| 0 := by reflexivity
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| (k+1) := proof ap (λX, Ω X) (homotopy_groups_mul3add2 k) qed
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/- The maps between the homotopy groups -/
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definition homotopy_groups_fun
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: Π(n : ℕ), homotopy_groups f (n+1) →* homotopy_groups f n
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| 0 := proof f qed
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| 1 := proof ppoint f qed
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| 2 := proof boundary_map f qed
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| 3 := proof ap1 f ∘* pinverse qed
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| 4 := proof ap1 (ppoint f) ∘* pinverse qed
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| 5 := proof ap1 (boundary_map f) ∘* pinverse qed
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| (k+6) := proof ap1 (ap1 (homotopy_groups_fun k)) qed
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definition homotopy_groups_fun_add6 [unfold_full] :
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homotopy_groups_fun f (n + 6) = ap1 (ap1 (homotopy_groups_fun f n)) :=
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proof idp qed
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/- this is a simpler defintion of the functions, but which are the same as the previous ones
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(there is a pointed homotopy) -/
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definition homotopy_groups_fun'
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: Π(n : ℕ), homotopy_groups f (n+1) →* homotopy_groups f n
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| 0 := proof f qed
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| 1 := proof ppoint f qed
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| 2 := proof boundary_map f qed
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| (k+3) := proof ap1 (homotopy_groups_fun' k) ∘* pinverse qed
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definition homotopy_groups_fun'_add3 [unfold_full] :
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homotopy_groups_fun' f (n+3) = ap1 (homotopy_groups_fun' f n) ∘* pinverse :=
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proof idp qed
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theorem homotopy_groups_fun_eq
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: Π(n : ℕ), homotopy_groups_fun f n ~* homotopy_groups_fun' f n
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| 0 := by reflexivity
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| 1 := by reflexivity
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| 2 := by reflexivity
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| 3 := by reflexivity
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| 4 := by reflexivity
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| 5 := by reflexivity
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| (k+6) :=
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begin
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rewrite [homotopy_groups_fun_add6 f k],
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replace (k + 6) with (k + 3 + 3),
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rewrite [homotopy_groups_fun'_add3 f (k+3)],
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rewrite [homotopy_groups_fun'_add3 f k],
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refine _ ⬝* pwhisker_right _ !ap1_compose⁻¹*,
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refine _ ⬝* !passoc⁻¹*,
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refine !comp_pid⁻¹* ⬝* _,
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refine pconcat2 _ _,
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/- Currently ap1_phomotopy is defined using function extensionality -/
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{ apply ap1_phomotopy, apply pap ap1, apply homotopy_groups_fun_eq},
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{ refine _ ⬝* (pwhisker_right _ ap1_pinverse)⁻¹*, fapply phomotopy.mk,
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{ intro q, esimp, exact !inv_inv⁻¹},
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{ reflexivity}}
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end
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definition homotopy_groups_fun_add3 :
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homotopy_groups_fun f (n + 3) ~* ap1 (homotopy_groups_fun f n) ∘* pinverse :=
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begin
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refine homotopy_groups_fun_eq f (n+3) ⬝* _,
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exact pwhisker_right _ (ap1_phomotopy (homotopy_groups_fun_eq f n)⁻¹*),
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end
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definition fiber_sequence_pequiv_homotopy_groups :
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Πn, fiber_sequence_carrier f n ≃* homotopy_groups f n
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| 0 := by reflexivity
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| 1 := by reflexivity
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| 2 := by reflexivity
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| (k+3) :=
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begin
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refine fiber_sequence_carrier_pequiv f k ⬝e* _,
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apply loop_space_pequiv,
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exact fiber_sequence_pequiv_homotopy_groups k
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end
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definition fiber_sequence_pequiv_homotopy_groups_add3
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: fiber_sequence_pequiv_homotopy_groups f (n + 3) =
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ap1 (fiber_sequence_pequiv_homotopy_groups f n) ∘* fiber_sequence_carrier_pequiv f n :=
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by reflexivity
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definition fiber_sequence_pequiv_homotopy_groups_3_phomotopy
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: fiber_sequence_pequiv_homotopy_groups f 3 ~* fiber_sequence_carrier_pequiv f 0 :=
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begin
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refine fiber_sequence_pequiv_homotopy_groups_add3 f 0 ⬝p* _,
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refine pwhisker_right _ ap1_id ⬝* _,
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apply pid_comp
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end
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theorem fiber_sequence_phomotopy_homotopy_groups' :
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Π(n : ℕ),
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fiber_sequence_pequiv_homotopy_groups f n ∘* fiber_sequence_fun f n ~*
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homotopy_groups_fun' f n ∘* fiber_sequence_pequiv_homotopy_groups f (n + 1)
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| 0 := by reflexivity
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| 1 := by reflexivity
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| 2 :=
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begin
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refine !pid_comp ⬝* _,
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replace homotopy_groups_fun' f 2 with boundary_map f,
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refine _ ⬝* pwhisker_left _ (fiber_sequence_pequiv_homotopy_groups_3_phomotopy f)⁻¹*,
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apply phomotopy_of_pinv_right_phomotopy,
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reflexivity
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end
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| (k+3) :=
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begin
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replace (k + 3 + 1) with (k + 1 + 3),
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rewrite [fiber_sequence_pequiv_homotopy_groups_add3 f k,
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fiber_sequence_pequiv_homotopy_groups_add3 f (k+1)],
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refine !passoc ⬝* _,
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refine pwhisker_left _ (fiber_sequence_fun_phomotopy f k) ⬝* _,
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
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apply pwhisker_right,
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rewrite [homotopy_groups_fun'_add3],
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refine _ ⬝* !passoc⁻¹*,
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refine _ ⬝* pwhisker_left _ !ap1_compose_pinverse,
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refine !passoc⁻¹* ⬝* _ ⬝* !passoc,
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apply pwhisker_right,
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refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose,
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apply ap1_phomotopy,
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exact fiber_sequence_phomotopy_homotopy_groups' k
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end
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theorem fiber_sequence_phomotopy_homotopy_groups (n : ℕ)
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(x : fiber_sequence_carrier f (n + 1)) :
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fiber_sequence_pequiv_homotopy_groups f n (fiber_sequence_fun f n x) =
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homotopy_groups_fun f n (fiber_sequence_pequiv_homotopy_groups f (n + 1) x) :=
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begin
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refine fiber_sequence_phomotopy_homotopy_groups' f n x ⬝ _,
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exact (homotopy_groups_fun_eq f n _)⁻¹
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end
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definition type_LES_of_homotopy_groups [constructor] : type_chain_complex +ℕ :=
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transfer_type_chain_complex
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(fiber_sequence f)
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(homotopy_groups_fun f)
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(fiber_sequence_pequiv_homotopy_groups f)
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(fiber_sequence_phomotopy_homotopy_groups f)
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definition is_exact_type_LES_of_homotopy_groups : is_exact_t (type_LES_of_homotopy_groups f) :=
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begin
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intro n,
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apply is_exact_at_t_transfer,
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apply is_exact_fiber_sequence
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end
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/- the long exact sequence of homotopy groups -/
|
||
definition LES_of_homotopy_groups [constructor] : chain_complex +ℕ :=
|
||
trunc_chain_complex
|
||
(transfer_type_chain_complex
|
||
(fiber_sequence f)
|
||
(homotopy_groups_fun f)
|
||
(fiber_sequence_pequiv_homotopy_groups f)
|
||
(fiber_sequence_phomotopy_homotopy_groups f))
|
||
|
||
/- the fiber sequence is exact -/
|
||
definition is_exact_LES_of_homotopy_groups : is_exact (LES_of_homotopy_groups f) :=
|
||
begin
|
||
intro n,
|
||
apply is_exact_at_trunc,
|
||
apply is_exact_type_LES_of_homotopy_groups
|
||
end
|
||
|
||
/- for a numeral, the carrier of the fiber sequence is definitionally what we want
|
||
(as pointed sets) -/
|
||
example : LES_of_homotopy_groups f 6 = π*[2] Y :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups f 7 = π*[2] X :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups f 8 = π*[2] (pfiber f) :> Set* := by reflexivity
|
||
|
||
/- for a numeral, the functions of the fiber sequence is definitionally what we want
|
||
(as pointed function). All these functions have at most one "pinverse" in them, and these
|
||
inverses are inside the π→*[2*k].
|
||
-/
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 6 = π→*[2] f
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 7 = π→*[2] (ppoint f)
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 8 = π→*[2] (boundary_map f)
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 9 = π→*[2] (ap1 f ∘* pinverse)
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 10 = π→*[2] (ap1 (ppoint f) ∘* pinverse)
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 11 = π→*[2] (ap1 (boundary_map f) ∘* pinverse)
|
||
:> (_ →* _) := by reflexivity
|
||
example : cc_to_fn (LES_of_homotopy_groups f) 12 = π→*[4] f
|
||
:> (_ →* _) := by reflexivity
|
||
|
||
/- the carrier of the fiber sequence is what we want for natural numbers of the form
|
||
3n, 3n+1 and 3n+2 -/
|
||
definition LES_of_homotopy_groups_mul3 (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n) = π*[n] Y :> Set* :=
|
||
begin
|
||
apply ptrunctype_eq_of_pType_eq,
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3 f n)
|
||
end
|
||
|
||
definition LES_of_homotopy_groups_mul3add1 (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n + 1) = π*[n] X :> Set* :=
|
||
begin
|
||
apply ptrunctype_eq_of_pType_eq,
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3add1 f n)
|
||
end
|
||
|
||
definition LES_of_homotopy_groups_mul3add2 (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n + 2) = π*[n] (pfiber f) :> Set* :=
|
||
begin
|
||
apply ptrunctype_eq_of_pType_eq,
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3add2 f n)
|
||
end
|
||
|
||
definition LES_of_homotopy_groups_mul3' (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n) = π*[n] Y :> Type :=
|
||
begin
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3 f n)
|
||
end
|
||
|
||
definition LES_of_homotopy_groups_mul3add1' (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n + 1) = π*[n] X :> Type :=
|
||
begin
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3add1 f n)
|
||
end
|
||
|
||
definition LES_of_homotopy_groups_mul3add2' (n : ℕ)
|
||
: LES_of_homotopy_groups f (3 * n + 2) = π*[n] (pfiber f) :> Type :=
|
||
begin
|
||
exact ap (ptrunc 0) (homotopy_groups_mul3add2 f n)
|
||
end
|
||
|
||
definition group_LES_of_homotopy_groups (n : ℕ) : group (LES_of_homotopy_groups f (n + 3)) :=
|
||
group_homotopy_group 0 (homotopy_groups f n)
|
||
|
||
definition comm_group_LES_of_homotopy_groups (n : ℕ) : comm_group (LES_of_homotopy_groups f (n + 6)) :=
|
||
comm_group_homotopy_group 0 (homotopy_groups f n)
|
||
|
||
end chain_complex
|
||
|
||
open group prod succ_str fin
|
||
|
||
/--------------
|
||
PART 3
|
||
--------------/
|
||
|
||
namespace chain_complex
|
||
|
||
--TODO: move
|
||
definition tr_mul_tr {A : Type*} (n : ℕ) (p q : Ω[n + 1] A) :
|
||
tr p *[πg[n+1] A] tr q = tr (p ⬝ q) :=
|
||
by reflexivity
|
||
|
||
definition is_homomorphism_cast_loop_space_succ_eq_in {A : Type*} (n : ℕ) :
|
||
is_homomorphism
|
||
(cast (ap (trunc 0 ∘ pointed.carrier) (loop_space_succ_eq_in A (succ n)))
|
||
: πg[n+1+1] A → πg[n+1] Ω A) :=
|
||
begin
|
||
intro g h, induction g with g, induction h with h,
|
||
xrewrite [tr_mul_tr, - + fn_cast_eq_cast_fn _ (λn, tr), tr_mul_tr, ↑cast, -tr_compose,
|
||
loop_space_succ_eq_in_concat, - + tr_compose],
|
||
end
|
||
|
||
definition is_homomorphism_inverse (A : Type*) (n : ℕ)
|
||
: is_homomorphism (λp, p⁻¹ : πag[n+2] A → πag[n+2] A) :=
|
||
begin
|
||
intro g h, rewrite mul.comm,
|
||
induction g with g, induction h with h,
|
||
exact ap tr !con_inv
|
||
end
|
||
|
||
section
|
||
universe variable u
|
||
parameters {X Y : pType.{u}} (f : X →* Y)
|
||
|
||
definition homotopy_groups2 [reducible] : +6ℕ → Type*
|
||
| (n, fin.mk 0 H) := Ω[2*n] Y
|
||
| (n, fin.mk 1 H) := Ω[2*n] X
|
||
| (n, fin.mk 2 H) := Ω[2*n] (pfiber f)
|
||
| (n, fin.mk 3 H) := Ω[2*n + 1] Y
|
||
| (n, fin.mk 4 H) := Ω[2*n + 1] X
|
||
| (n, fin.mk k H) := Ω[2*n + 1] (pfiber f)
|
||
|
||
definition homotopy_groups2_add1 (n : ℕ) : Π(x : fin (succ 5)),
|
||
homotopy_groups2 (n+1, x) = Ω Ω(homotopy_groups2 (n, x))
|
||
| (fin.mk 0 H) := by reflexivity
|
||
| (fin.mk 1 H) := by reflexivity
|
||
| (fin.mk 2 H) := by reflexivity
|
||
| (fin.mk 3 H) := by reflexivity
|
||
| (fin.mk 4 H) := by reflexivity
|
||
| (fin.mk 5 H) := by reflexivity
|
||
| (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition homotopy_groups_fun2 : Π(n : +6ℕ), homotopy_groups2 (S n) →* homotopy_groups2 n
|
||
| (n, fin.mk 0 H) := proof Ω→[2*n] f qed
|
||
| (n, fin.mk 1 H) := proof Ω→[2*n] (ppoint f) qed
|
||
| (n, fin.mk 2 H) :=
|
||
proof Ω→[2*n] (boundary_map f) ∘* pcast (loop_space_succ_eq_in Y (2*n)) qed
|
||
| (n, fin.mk 3 H) := proof Ω→[2*n + 1] f ∘* pinverse qed
|
||
| (n, fin.mk 4 H) := proof Ω→[2*n + 1] (ppoint f) ∘* pinverse qed
|
||
| (n, fin.mk 5 H) :=
|
||
proof (Ω→[2*n + 1] (boundary_map f) ∘* pinverse) ∘* pcast (loop_space_succ_eq_in Y (2*n+1)) qed
|
||
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition homotopy_groups_fun2_add1_0 (n : ℕ) (H : 0 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 0 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 0 H))) :=
|
||
by reflexivity
|
||
|
||
definition homotopy_groups_fun2_add1_1 (n : ℕ) (H : 1 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 1 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 1 H))) :=
|
||
by reflexivity
|
||
|
||
definition homotopy_groups_fun2_add1_2 (n : ℕ) (H : 2 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 2 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 2 H))) :=
|
||
begin
|
||
esimp, refine _ ⬝* (ap1_phomotopy !ap1_compose)⁻¹*, refine _ ⬝* !ap1_compose⁻¹*,
|
||
apply pwhisker_left,
|
||
refine !pcast_ap_loop_space ⬝* ap1_phomotopy !pcast_ap_loop_space,
|
||
end
|
||
|
||
definition homotopy_groups_fun2_add1_3 (n : ℕ) (H : 3 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 3 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 3 H))) :=
|
||
begin
|
||
esimp, refine _ ⬝* (ap1_phomotopy !ap1_compose)⁻¹*, refine _ ⬝* !ap1_compose⁻¹*,
|
||
apply pwhisker_left,
|
||
exact ap1_pinverse⁻¹* ⬝* ap1_phomotopy !ap1_pinverse⁻¹*
|
||
end
|
||
|
||
definition homotopy_groups_fun2_add1_4 (n : ℕ) (H : 4 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 4 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 4 H))) :=
|
||
begin
|
||
esimp, refine _ ⬝* (ap1_phomotopy !ap1_compose)⁻¹*, refine _ ⬝* !ap1_compose⁻¹*,
|
||
apply pwhisker_left,
|
||
exact ap1_pinverse⁻¹* ⬝* ap1_phomotopy !ap1_pinverse⁻¹*
|
||
end
|
||
|
||
definition homotopy_groups_fun2_add1_5 (n : ℕ) (H : 5 < succ 5)
|
||
: homotopy_groups_fun2 (n+1, fin.mk 5 H) ~*
|
||
cast proof idp qed ap1 (ap1 (homotopy_groups_fun2 (n, fin.mk 5 H))) :=
|
||
begin
|
||
esimp, refine _ ⬝* (ap1_phomotopy !ap1_compose)⁻¹*, refine _ ⬝* !ap1_compose⁻¹*,
|
||
apply pconcat2,
|
||
{ esimp, refine _ ⬝* (ap1_phomotopy !ap1_compose)⁻¹*, refine _ ⬝* !ap1_compose⁻¹*,
|
||
apply pwhisker_left,
|
||
exact ap1_pinverse⁻¹* ⬝* ap1_phomotopy !ap1_pinverse⁻¹*},
|
||
{ refine !pcast_ap_loop_space ⬝* ap1_phomotopy !pcast_ap_loop_space}
|
||
end
|
||
|
||
definition nat_of_str [unfold 2] [reducible] {n : ℕ} : ℕ × fin (succ n) → ℕ :=
|
||
λx, succ n * pr1 x + val (pr2 x)
|
||
|
||
definition str_of_nat {n : ℕ} : ℕ → ℕ × fin (succ n) :=
|
||
λm, (m / (succ n), mk_mod n m)
|
||
|
||
definition nat_of_str_6S [unfold 2] [reducible]
|
||
: Π(x : stratified +ℕ 5), nat_of_str x + 1 = nat_of_str (@S (stratified +ℕ 5) x)
|
||
| (n, fin.mk 0 H) := by reflexivity
|
||
| (n, fin.mk 1 H) := by reflexivity
|
||
| (n, fin.mk 2 H) := by reflexivity
|
||
| (n, fin.mk 3 H) := by reflexivity
|
||
| (n, fin.mk 4 H) := by reflexivity
|
||
| (n, fin.mk 5 H) := by reflexivity
|
||
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition fin_prod_nat_equiv_nat [constructor] (n : ℕ) : ℕ × fin (succ n) ≃ ℕ :=
|
||
equiv.MK nat_of_str str_of_nat
|
||
abstract begin
|
||
intro m, unfold [nat_of_str, str_of_nat, mk_mod],
|
||
refine _ ⬝ (eq_div_mul_add_mod m (succ n))⁻¹,
|
||
rewrite [mul.comm]
|
||
end end
|
||
abstract begin
|
||
intro x, cases x with m k,
|
||
cases k with k H,
|
||
apply prod_eq: esimp [str_of_nat],
|
||
{ rewrite [add.comm, add_mul_div_self_left _ _ (!zero_lt_succ),
|
||
div_eq_zero_of_lt H, zero_add]},
|
||
{ apply eq_of_veq, esimp [mk_mod],
|
||
rewrite [add.comm, add_mul_mod_self_left, mod_eq_of_lt H]}
|
||
end end
|
||
|
||
/-
|
||
note: in the following theorem the (n+1) case is 6 times the same,
|
||
so maybe this can be simplified
|
||
-/
|
||
definition homotopy_groups2_pequiv' : Π(n : ℕ) (x : fin (nat.succ 5)),
|
||
homotopy_groups f (nat_of_str (n, x)) ≃* homotopy_groups2 (n, x)
|
||
| 0 (fin.mk 0 H) := by reflexivity
|
||
| 0 (fin.mk 1 H) := by reflexivity
|
||
| 0 (fin.mk 2 H) := by reflexivity
|
||
| 0 (fin.mk 3 H) := by reflexivity
|
||
| 0 (fin.mk 4 H) := by reflexivity
|
||
| 0 (fin.mk 5 H) := by reflexivity
|
||
| (n+1) (fin.mk 0 H) :=
|
||
begin
|
||
-- uncomment the next two lines to have prettier subgoals
|
||
-- esimp, replace (succ 5 * (n + 1) + 0) with (6*n+3+3),
|
||
-- rewrite [+homotopy_groups_add3, homotopy_groups2_add1],
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 0 H)
|
||
end
|
||
| (n+1) (fin.mk 1 H) :=
|
||
begin
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 1 H)
|
||
end
|
||
| (n+1) (fin.mk 2 H) :=
|
||
begin
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 2 H)
|
||
end
|
||
| (n+1) (fin.mk 3 H) :=
|
||
begin
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 3 H)
|
||
end
|
||
| (n+1) (fin.mk 4 H) :=
|
||
begin
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 4 H)
|
||
end
|
||
| (n+1) (fin.mk 5 H) :=
|
||
begin
|
||
apply loop_space_pequiv, apply loop_space_pequiv,
|
||
rexact homotopy_groups2_pequiv' n (fin.mk 5 H)
|
||
end
|
||
| n (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition homotopy_groups2_pequiv : Π(x : +6ℕ),
|
||
homotopy_groups f (nat_of_str x) ≃* homotopy_groups2 x
|
||
| (n, x) := homotopy_groups2_pequiv' n x
|
||
|
||
/- all cases where n>0 are basically the same -/
|
||
definition homotopy_groups_fun2_phomotopy (x : +6ℕ) :
|
||
homotopy_groups2_pequiv x ∘* homotopy_groups_fun f (nat_of_str x) ~*
|
||
(homotopy_groups_fun2 x ∘* homotopy_groups2_pequiv (S x))
|
||
∘* pcast (ap (homotopy_groups f) (nat_of_str_6S x)) :=
|
||
begin
|
||
cases x with n x, cases x with k H,
|
||
cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 1,
|
||
cases k with k, rotate 1, cases k with k, rotate 1, cases k with k, rotate 2,
|
||
{ /-k=0-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
|
||
reflexivity},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_0)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=1-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
|
||
reflexivity},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_1)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=2-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* !comp_pid⁻¹*,
|
||
reflexivity},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_2)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=3-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
|
||
reflexivity},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_3)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=4-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹* ⬝* !comp_pid⁻¹*,
|
||
reflexivity},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_4)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=5-/
|
||
induction n with n IH,
|
||
{ refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
|
||
refine !comp_pid⁻¹* ⬝* pconcat2 _ _,
|
||
{ exact !comp_pid⁻¹*},
|
||
{ refine cast (ap (λx, _ ~* loop_space_pequiv x) !loop_space_pequiv_rfl)⁻¹ _,
|
||
refine cast (ap (λx, _ ~* x) !loop_space_pequiv_rfl)⁻¹ _, reflexivity}},
|
||
{ refine _ ⬝* !comp_pid⁻¹*,
|
||
refine _ ⬝* pwhisker_right _ (!homotopy_groups_fun2_add1_5)⁻¹*,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
refine !ap1_compose⁻¹* ⬝* _ ⬝* !ap1_compose, apply ap1_phomotopy,
|
||
exact IH ⬝* !comp_pid}},
|
||
{ /-k=k'+6-/ exfalso, apply lt_le_antisymm H, apply le_add_left}
|
||
end
|
||
|
||
definition type_LES_of_homotopy_groups2 [constructor] : type_chain_complex +6ℕ :=
|
||
transfer_type_chain_complex2
|
||
(type_LES_of_homotopy_groups f)
|
||
!fin_prod_nat_equiv_nat
|
||
nat_of_str_6S
|
||
@homotopy_groups_fun2
|
||
@homotopy_groups2_pequiv
|
||
begin
|
||
intro m x,
|
||
refine homotopy_groups_fun2_phomotopy m x ⬝ _,
|
||
apply ap (homotopy_groups_fun2 m), apply ap (homotopy_groups2_pequiv (S m)),
|
||
esimp, exact ap010 cast !ap_compose⁻¹ x
|
||
end
|
||
|
||
definition is_exact_type_LES_of_homotopy_groups2 : is_exact_t (type_LES_of_homotopy_groups2) :=
|
||
begin
|
||
intro n,
|
||
apply is_exact_at_transfer2,
|
||
apply is_exact_type_LES_of_homotopy_groups
|
||
end
|
||
|
||
definition LES_of_homotopy_groups2 [constructor] : chain_complex +6ℕ :=
|
||
trunc_chain_complex type_LES_of_homotopy_groups2
|
||
|
||
/--------------
|
||
PART 4
|
||
--------------/
|
||
|
||
definition homotopy_groups3 [reducible] : +6ℕ → Set*
|
||
| (n, fin.mk 0 H) := π*[2*n] Y
|
||
| (n, fin.mk 1 H) := π*[2*n] X
|
||
| (n, fin.mk 2 H) := π*[2*n] (pfiber f)
|
||
| (n, fin.mk 3 H) := π*[2*n + 1] Y
|
||
| (n, fin.mk 4 H) := π*[2*n + 1] X
|
||
| (n, fin.mk k H) := π*[2*n + 1] (pfiber f)
|
||
|
||
definition homotopy_groups3eq2 [reducible]
|
||
: Π(n : +6ℕ), ptrunc 0 (homotopy_groups2 n) ≃* homotopy_groups3 n
|
||
| (n, fin.mk 0 H) := by reflexivity
|
||
| (n, fin.mk 1 H) := by reflexivity
|
||
| (n, fin.mk 2 H) := by reflexivity
|
||
| (n, fin.mk 3 H) := by reflexivity
|
||
| (n, fin.mk 4 H) := by reflexivity
|
||
| (n, fin.mk 5 H) := by reflexivity
|
||
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition homotopy_groups_fun3 : Π(n : +6ℕ), homotopy_groups3 (S n) →* homotopy_groups3 n
|
||
| (n, fin.mk 0 H) := proof π→*[2*n] f qed
|
||
| (n, fin.mk 1 H) := proof π→*[2*n] (ppoint f) qed
|
||
| (n, fin.mk 2 H) :=
|
||
proof π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) qed
|
||
| (n, fin.mk 3 H) := proof π→*[2*n + 1] f ∘* tinverse qed
|
||
| (n, fin.mk 4 H) := proof π→*[2*n + 1] (ppoint f) ∘* tinverse qed
|
||
| (n, fin.mk 5 H) :=
|
||
proof (π→*[2*n + 1] (boundary_map f) ∘* tinverse)
|
||
∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) qed
|
||
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition homotopy_groups_fun3eq2 [reducible]
|
||
: Π(n : +6ℕ), homotopy_groups3eq2 n ∘* ptrunc_functor 0 (homotopy_groups_fun2 n) ~*
|
||
homotopy_groups_fun3 n ∘* homotopy_groups3eq2 (S n)
|
||
| (n, fin.mk 0 H) := by reflexivity
|
||
| (n, fin.mk 1 H) := by reflexivity
|
||
| (n, fin.mk 2 H) :=
|
||
begin
|
||
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
|
||
refine !ptrunc_functor_pcompose ⬝* _,
|
||
apply pwhisker_left, apply ptrunc_functor_pcast,
|
||
end
|
||
| (n, fin.mk 3 H) :=
|
||
begin
|
||
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
|
||
refine !ptrunc_functor_pcompose ⬝* _,
|
||
apply pwhisker_left, apply ptrunc_functor_pinverse
|
||
end
|
||
| (n, fin.mk 4 H) :=
|
||
begin
|
||
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
|
||
refine !ptrunc_functor_pcompose ⬝* _,
|
||
apply pwhisker_left, apply ptrunc_functor_pinverse
|
||
end
|
||
| (n, fin.mk 5 H) :=
|
||
begin
|
||
refine !pid_comp ⬝* _ ⬝* !comp_pid⁻¹*,
|
||
refine !ptrunc_functor_pcompose ⬝* _,
|
||
apply pconcat2,
|
||
{ refine !ptrunc_functor_pcompose ⬝* _,
|
||
apply pwhisker_left, apply ptrunc_functor_pinverse},
|
||
{ apply ptrunc_functor_pcast}
|
||
end
|
||
| (n, fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition LES_of_homotopy_groups3 [constructor] : chain_complex +6ℕ :=
|
||
transfer_chain_complex
|
||
LES_of_homotopy_groups2
|
||
homotopy_groups_fun3
|
||
homotopy_groups3eq2
|
||
homotopy_groups_fun3eq2
|
||
|
||
definition is_exact_LES_of_homotopy_groups3 : is_exact (LES_of_homotopy_groups3) :=
|
||
begin
|
||
intro n,
|
||
apply is_exact_at_transfer,
|
||
apply is_exact_at_trunc,
|
||
apply is_exact_type_LES_of_homotopy_groups2
|
||
end
|
||
|
||
end
|
||
|
||
open is_trunc
|
||
universe variable u
|
||
variables {X Y : pType.{u}} (f : X →* Y) (n : ℕ)
|
||
include f
|
||
|
||
/- the carrier of the fiber sequence is definitionally what we want (as pointed sets) -/
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 6) = π*[2] Y :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 7) = π*[2] X :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 8) = π*[2] (pfiber f) :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 9) = π*[3] Y :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 10) = π*[3] X :> Set* := by reflexivity
|
||
example : LES_of_homotopy_groups3 f (str_of_nat 11) = π*[3] (pfiber f) :> Set* := by reflexivity
|
||
|
||
definition LES_of_homotopy_groups3_0 : LES_of_homotopy_groups3 f (n, 0) = π*[2*n] Y :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups3_1 : LES_of_homotopy_groups3 f (n, 1) = π*[2*n] X :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups3_2 : LES_of_homotopy_groups3 f (n, 2) = π*[2*n] (pfiber f) :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups3_3 : LES_of_homotopy_groups3 f (n, 3) = π*[2*n + 1] Y :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups3_4 : LES_of_homotopy_groups3 f (n, 4) = π*[2*n + 1] X :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups3_5 : LES_of_homotopy_groups3 f (n, 5) = π*[2*n + 1] (pfiber f):=
|
||
by reflexivity
|
||
|
||
/- the functions of the fiber sequence is definitionally what we want (as pointed function).
|
||
-/
|
||
|
||
definition LES_of_homotopy_groups_fun3_0 :
|
||
cc_to_fn (LES_of_homotopy_groups3 f) (n, 0) = π→*[2*n] f :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups_fun3_1 :
|
||
cc_to_fn (LES_of_homotopy_groups3 f) (n, 1) = π→*[2*n] (ppoint f) :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups_fun3_2 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 2) =
|
||
π→*[2*n] (boundary_map f) ∘* pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n))) :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups_fun3_3 :
|
||
cc_to_fn (LES_of_homotopy_groups3 f) (n, 3) = π→*[2*n + 1] f ∘* tinverse :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups_fun3_4 :
|
||
cc_to_fn (LES_of_homotopy_groups3 f) (n, 4) = π→*[2*n + 1] (ppoint f) ∘* tinverse :=
|
||
by reflexivity
|
||
definition LES_of_homotopy_groups_fun3_5 : cc_to_fn (LES_of_homotopy_groups3 f) (n, 5) =
|
||
(π→*[2*n + 1] (boundary_map f) ∘* tinverse) ∘*
|
||
pcast (ap (ptrunc 0) (loop_space_succ_eq_in Y (2*n+1))) :=
|
||
by reflexivity
|
||
|
||
definition group_LES_of_homotopy_groups3_0 :
|
||
Π(k : ℕ) (H : k + 3 < succ 5), group (LES_of_homotopy_groups3 f (0, fin.mk (k+3) H))
|
||
| 0 H := begin rexact group_homotopy_group 0 Y end
|
||
| 1 H := begin rexact group_homotopy_group 0 X end
|
||
| 2 H := begin rexact group_homotopy_group 0 (pfiber f) end
|
||
| (k+3) H := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition comm_group_LES_of_homotopy_groups3 (n : ℕ) : Π(x : fin (succ 5)),
|
||
comm_group (LES_of_homotopy_groups3 f (n + 1, x))
|
||
| (fin.mk 0 H) := proof comm_group_homotopy_group (2*n) Y qed
|
||
| (fin.mk 1 H) := proof comm_group_homotopy_group (2*n) X qed
|
||
| (fin.mk 2 H) := proof comm_group_homotopy_group (2*n) (pfiber f) qed
|
||
| (fin.mk 3 H) := proof comm_group_homotopy_group (2*n+1) Y qed
|
||
| (fin.mk 4 H) := proof comm_group_homotopy_group (2*n+1) X qed
|
||
| (fin.mk 5 H) := proof comm_group_homotopy_group (2*n+1) (pfiber f) qed
|
||
| (fin.mk (k+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
definition CommGroup_LES_of_homotopy_groups3 (n : +6ℕ) : CommGroup.{u} :=
|
||
CommGroup.mk (LES_of_homotopy_groups3 f (pr1 n + 1, pr2 n))
|
||
(comm_group_LES_of_homotopy_groups3 f (pr1 n) (pr2 n))
|
||
|
||
definition homomorphism_LES_of_homotopy_groups_fun3 : Π(k : +6ℕ),
|
||
CommGroup_LES_of_homotopy_groups3 f (S k) →g CommGroup_LES_of_homotopy_groups3 f k
|
||
| (k, fin.mk 0 H) :=
|
||
proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 0))
|
||
(phomotopy_group_functor_mul _ _) qed
|
||
| (k, fin.mk 1 H) :=
|
||
proof homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 1))
|
||
(phomotopy_group_functor_mul _ _) qed
|
||
| (k, fin.mk 2 H) :=
|
||
begin
|
||
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 2)),
|
||
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_2],
|
||
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1)] boundary_map f) _ _ _,
|
||
{ apply group_homotopy_group ((2 * k) + 1)},
|
||
{ apply phomotopy_group_functor_mul},
|
||
{ rewrite [▸*, -ap_compose', ▸*],
|
||
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
|
||
end
|
||
| (k, fin.mk 3 H) :=
|
||
begin
|
||
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 3)),
|
||
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_3],
|
||
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] f) tinverse _ _,
|
||
{ apply group_homotopy_group (2 * (k+1))},
|
||
{ apply phomotopy_group_functor_mul},
|
||
{ apply is_homomorphism_inverse} end end
|
||
end
|
||
| (k, fin.mk 4 H) :=
|
||
begin
|
||
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 4)),
|
||
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_4],
|
||
refine @is_homomorphism_compose _ _ _ _ _ _ (π→*[2 * (k + 1) + 1] (ppoint f)) tinverse _ _,
|
||
{ apply group_homotopy_group (2 * (k+1))},
|
||
{ apply phomotopy_group_functor_mul},
|
||
{ apply is_homomorphism_inverse} end end
|
||
end
|
||
| (k, fin.mk 5 H) :=
|
||
begin
|
||
apply homomorphism.mk (cc_to_fn (LES_of_homotopy_groups3 f) (k + 1, 5)),
|
||
exact abstract begin rewrite [LES_of_homotopy_groups_fun3_5],
|
||
refine @is_homomorphism_compose _ _ _ _ _ _
|
||
(π→*[2 * (k + 1) + 1] (boundary_map f) ∘ tinverse) _ _ _,
|
||
{ refine @is_homomorphism_compose _ _ _ _ _ _
|
||
(π→*[2 * (k + 1) + 1] (boundary_map f)) tinverse _ _,
|
||
{ apply group_homotopy_group (2 * (k+1))},
|
||
{ apply phomotopy_group_functor_mul},
|
||
{ apply is_homomorphism_inverse}},
|
||
{ rewrite [▸*, -ap_compose', ▸*],
|
||
apply is_homomorphism_cast_loop_space_succ_eq_in} end end
|
||
end
|
||
| (k, fin.mk (l+6) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
|
||
|
||
--TODO: the maps 3, 4 and 5 are anti-homomorphisms.
|
||
|
||
end chain_complex
|