100 lines
3.7 KiB
Text
100 lines
3.7 KiB
Text
import homotopy.sphere2 ..move_to_lib
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open fin eq equiv group algebra sphere.ops pointed nat int trunc is_equiv function circle
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definition eq_one_or_eq_neg_one_of_mul_eq_one {n : ℤ} (m : ℤ) (p : n * m = 1) : n = 1 ⊎ n = -1 :=
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sorry
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definition endomorphism_int_unbundled (f : ℤ → ℤ) [is_add_homomorphism f] (n : ℤ) :
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f n = f 1 * n :=
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begin
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induction n using rec_nat_on with n IH n IH,
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{ refine respect_zero f ⬝ _, exact !mul_zero⁻¹ },
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{ refine respect_add f n 1 ⬝ _, rewrite IH,
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rewrite [↑int.succ, left_distrib], apply ap (λx, _ + x), exact !mul_one⁻¹},
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{ rewrite [neg_nat_succ], refine respect_add f (-n) (- 1) ⬝ _,
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rewrite [IH, ↑int.pred, mul_sub_left_distrib], apply ap (λx, _ + x),
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refine _ ⬝ ap neg !mul_one⁻¹, exact respect_neg f 1 }
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end
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namespace sphere
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attribute fundamental_group_of_circle fg_carrier_equiv_int [constructor]
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attribute untrunc_of_is_trunc [unfold 4]
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definition surf_eq_loop : @surf 1 = circle.loop := sorry
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-- definition π2S2_surf : π2S2 (tr surf) = 1 :> ℤ :=
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-- begin
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-- unfold [π2S2, chain_complex.LES_of_homotopy_groups],
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-- end
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-- check (pmap.to_fun
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-- (chain_complex.cc_to_fn
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-- (chain_complex.LES_of_homotopy_groups
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-- hopf.complex_phopf)
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-- (pair 1 2))
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-- (tr surf))
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-- eval (pmap.to_fun
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-- (chain_complex.cc_to_fn
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-- (chain_complex.LES_of_homotopy_groups
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-- hopf.complex_phopf)
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-- (pair 1 2))
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-- (tr surf))
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definition πnSn_surf (n : ℕ) : πnSn n (tr surf) = 1 :> ℤ :=
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begin
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cases n with n IH,
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{ refine ap (πnSn _ ∘ tr) surf_eq_loop ⬝ _, apply transport_code_loop },
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{ unfold [πnSn], exact sorry}
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end
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definition deg {n : ℕ} [H : is_succ n] (f : S* n →* S* n) : ℤ :=
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by induction H with n; exact πnSn n ((π→g[n+1] f) (tr surf))
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definition deg_id (n : ℕ) [H : is_succ n] : deg (pid (S* n)) = (1 : ℤ) :=
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by induction H with n;
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exact ap (πnSn n) (phomotopy_group_functor_pid (succ n) (S* (succ n)) (tr surf)) ⬝ πnSn_surf n
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definition deg_phomotopy {n : ℕ} [H : is_succ n] {f g : S* n →* S* n} (p : f ~* g) :
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deg f = deg g :=
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begin
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induction H with n,
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exact ap (πnSn n) (homotopy_group_functor_phomotopy (succ n) p (tr surf)),
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end
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definition endomorphism_int (f : gℤ →g gℤ) (n : ℤ) : f n = f (1 : ℤ) *[ℤ] n :=
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@endomorphism_int_unbundled f (homomorphism.addstruct f) n
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definition endomorphism_equiv_Z {X : Group} (e : X ≃g gℤ) {one : X}
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(p : e one = 1 :> ℤ) (φ : X →g X) (x : X) : e (φ x) = e (φ one) *[ℤ] e x :=
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begin
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revert x, refine equiv_rect' (equiv_of_isomorphism e) _ _,
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intro n,
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refine endomorphism_int (e ∘g φ ∘g e⁻¹ᵍ) n ⬝ _,
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refine ap011 (@mul ℤ _) _ _,
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{ esimp, apply ap (e ∘ φ), refine ap e⁻¹ᵍ p⁻¹ ⬝ _,
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exact to_left_inv (equiv_of_isomorphism e) one },
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{ symmetry, exact to_right_inv (equiv_of_isomorphism e) n}
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end
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definition deg_compose {n : ℕ} [H : is_succ n] (f g : S* n →* S* n) :
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deg (g ∘* f) = deg g *[ℤ] deg f :=
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begin
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induction H with n,
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refine ap (πnSn n) (homotopy_group_functor_compose (succ n) g f (tr surf)) ⬝ _,
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apply endomorphism_equiv_Z !πnSn !πnSn_surf (π→g[n+1] g)
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end
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definition deg_equiv {n : ℕ} [H : is_succ n] (f : S* n ≃* S* n) :
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deg f = 1 ⊎ deg f = -1 :=
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begin
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induction H with n,
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apply eq_one_or_eq_neg_one_of_mul_eq_one (deg f⁻¹ᵉ*),
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refine !deg_compose⁻¹ ⬝ _,
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refine deg_phomotopy (pright_inv f) ⬝ _,
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apply deg_id,
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end
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end sphere
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