181 lines
6.8 KiB
Text
181 lines
6.8 KiB
Text
/-
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Copyright (c) 2015 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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homotopy groups of a pointed space
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-/
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import .trunc_group types.trunc .group_theory
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open nat eq pointed trunc is_trunc algebra group function equiv unit
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namespace eq
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definition phomotopy_group [reducible] [constructor] (n : ℕ) (A : Type*) : Set* :=
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ptrunc 0 (Ω[n] A)
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definition homotopy_group [reducible] (n : ℕ) (A : Type*) : Type :=
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phomotopy_group n A
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notation `π*[`:95 n:0 `] `:0 := phomotopy_group n
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notation `π[`:95 n:0 `] `:0 := homotopy_group n
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definition group_homotopy_group [instance] [constructor] [reducible] (n : ℕ) (A : Type*)
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: group (π[succ n] A) :=
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trunc_group concat inverse idp con.assoc idp_con con_idp con.left_inv
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definition comm_group_homotopy_group [constructor] [reducible] (n : ℕ) (A : Type*)
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: comm_group (π[succ (succ n)] A) :=
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trunc_comm_group concat inverse idp con.assoc idp_con con_idp con.left_inv eckmann_hilton
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local attribute comm_group_homotopy_group [instance]
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definition ghomotopy_group [constructor] (n : ℕ) (A : Type*) : Group :=
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Group.mk (π[succ n] A) _
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definition cghomotopy_group [constructor] (n : ℕ) (A : Type*) : CommGroup :=
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CommGroup.mk (π[succ (succ n)] A) _
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definition fundamental_group [constructor] (A : Type*) : Group :=
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ghomotopy_group zero A
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notation `πg[`:95 n:0 ` +1] `:0 A:95 := ghomotopy_group n A
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notation `πag[`:95 n:0 ` +2] `:0 A:95 := cghomotopy_group n A
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notation `π₁` := fundamental_group
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definition tr_mul_tr {n : ℕ} {A : Type*} (p q : Ω[n + 1] A) :
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tr p *[πg[n+1] A] tr q = tr (p ⬝ q) :=
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by reflexivity
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definition tr_mul_tr' {n : ℕ} {A : Type*} (p q : Ω[succ n] A)
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: tr p *[π[succ n] A] tr q = tr (p ⬝ q) :=
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idp
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definition phomotopy_group_pequiv [constructor] (n : ℕ) {A B : Type*} (H : A ≃* B)
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: π*[n] A ≃* π*[n] B :=
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ptrunc_pequiv_ptrunc 0 (loopn_pequiv_loopn n H)
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definition phomotopy_group_pequiv_loop_ptrunc [constructor] (k : ℕ) (A : Type*) :
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π*[k] A ≃* Ω[k] (ptrunc k A) :=
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begin
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refine !iterated_loop_ptrunc_pequiv⁻¹ᵉ* ⬝e* _,
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exact loopn_pequiv_loopn k (pequiv_of_eq begin rewrite [trunc_index.zero_add] end)
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end
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theorem trivial_homotopy_of_is_set (A : Type*) [H : is_set A] (n : ℕ) : πg[n+1] A ≃g G0 :=
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begin
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apply trivial_group_of_is_contr,
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apply is_trunc_trunc_of_is_trunc,
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apply is_contr_loop_of_is_trunc,
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apply is_trunc_succ_succ_of_is_set
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end
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definition phomotopy_group_succ_out (A : Type*) (n : ℕ) : π*[n + 1] A = π₁ Ω[n] A := idp
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definition phomotopy_group_succ_in (A : Type*) (n : ℕ) : π*[n + 1] A = π*[n] Ω A :=
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ap (ptrunc 0) (loop_space_succ_eq_in A n)
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definition ghomotopy_group_succ_out (A : Type*) (n : ℕ) : πg[n +1] A = π₁ Ω[n] A := idp
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definition ghomotopy_group_succ_in (A : Type*) (n : ℕ) : πg[succ n +1] A ≃g πg[n +1] Ω A :=
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begin
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fapply isomorphism_of_equiv,
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{ apply equiv_of_eq, exact ap (ptrunc 0) (loop_space_succ_eq_in A (succ n))},
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{ exact abstract [irreducible] begin refine trunc.rec _, intro p, refine trunc.rec _, intro q,
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rewrite [▸*,-+tr_eq_cast_ap, +trunc_transport], refine !trunc_transport ⬝ _, apply ap tr,
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apply loop_space_succ_eq_in_concat end end},
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end
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definition phomotopy_group_functor [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
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: π*[n] A →* π*[n] B :=
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ptrunc_functor 0 (apn n f)
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definition homotopy_group_functor (n : ℕ) {A B : Type*} (f : A →* B) : π[n] A → π[n] B :=
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phomotopy_group_functor n f
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notation `π→*[`:95 n:0 `] `:0 := phomotopy_group_functor n
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notation `π→[`:95 n:0 `] `:0 := homotopy_group_functor n
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definition tinverse [constructor] {X : Type*} : π*[1] X →* π*[1] X :=
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ptrunc_functor 0 pinverse
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definition is_equiv_tinverse [constructor] (A : Type*) : is_equiv (@tinverse A) :=
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by apply @is_equiv_trunc_functor; apply is_equiv_eq_inverse
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definition ptrunc_functor_pinverse [constructor] {X : Type*}
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: ptrunc_functor 0 (@pinverse X) ~* @tinverse X :=
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begin
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fapply phomotopy.mk,
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{ reflexivity},
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{ reflexivity}
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end
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definition phomotopy_group_functor_mul [constructor] (n : ℕ) {A B : Type*} (g : A →* B)
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(p q : πg[n+1] A) :
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(π→[n + 1] g) (p *[πg[n+1] A] q) = (π→[n + 1] g) p *[πg[n+1] B] (π→[n + 1] g) q :=
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begin
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unfold [ghomotopy_group, homotopy_group] at *,
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refine @trunc.rec _ _ _ (λq, !is_trunc_eq) _ p, clear p, intro p,
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refine @trunc.rec _ _ _ (λq, !is_trunc_eq) _ q, clear q, intro q,
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apply ap tr, apply apn_con
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end
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definition homotopy_group_homomorphism [constructor] (n : ℕ) {A B : Type*} (f : A →* B)
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: πg[n+1] A →g πg[n+1] B :=
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begin
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fconstructor,
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{ exact phomotopy_group_functor (n+1) f},
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{ apply phomotopy_group_functor_mul}
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end
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definition homotopy_group_isomorphism_of_pequiv [constructor] (n : ℕ) {A B : Type*} (f : A ≃* B)
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: πg[n+1] A ≃g πg[n+1] B :=
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begin
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apply isomorphism.mk (homotopy_group_homomorphism n f),
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esimp, apply is_equiv_trunc_functor, apply is_equiv_apn,
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end
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definition homotopy_group_add (A : Type*) (n m : ℕ) : πg[n+m +1] A ≃g πg[n +1] Ω[m] A :=
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begin
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revert A, induction m with m IH: intro A,
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{ reflexivity},
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{ esimp [iterated_ploop_space, nat.add], refine !ghomotopy_group_succ_in ⬝g _, refine !IH ⬝g _,
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apply homotopy_group_isomorphism_of_pequiv,
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exact pequiv_of_eq !loop_space_succ_eq_in⁻¹}
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end
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theorem trivial_homotopy_add_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ)
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(H : is_set (Ω[n] A)) : πg[m+n+1] A ≃g G0 :=
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!homotopy_group_add ⬝g !trivial_homotopy_of_is_set
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theorem trivial_homotopy_le_of_is_set_loop_space {A : Type*} {n : ℕ} (m : ℕ) (H1 : n ≤ m)
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(H2 : is_set (Ω[n] A)) : πg[m+1] A ≃g G0 :=
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obtain (k : ℕ) (p : n + k = m), from le.elim H1,
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isomorphism_of_eq (ap (λx, πg[x+1] A) (p⁻¹ ⬝ add.comm n k)) ⬝g
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trivial_homotopy_add_of_is_set_loop_space k H2
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/- some homomorphisms -/
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definition is_homomorphism_cast_loop_space_succ_eq_in {A : Type*} (n : ℕ) :
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is_homomorphism
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(cast (ap (trunc 0 ∘ pointed.carrier) (loop_space_succ_eq_in A (succ n)))
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: πg[n+1+1] A → πg[n+1] Ω A) :=
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begin
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intro g h, induction g with g, induction h with h,
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xrewrite [tr_mul_tr, - + fn_cast_eq_cast_fn _ (λn, tr), tr_mul_tr, ↑cast, -tr_compose,
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loop_space_succ_eq_in_concat, - + tr_compose],
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end
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definition is_homomorphism_inverse (A : Type*) (n : ℕ)
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: is_homomorphism (λp, p⁻¹ : πag[n+2] A → πag[n+2] A) :=
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begin
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intro g h, rewrite mul.comm,
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induction g with g, induction h with h,
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exact ap tr !con_inv
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end
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notation `π→g[`:95 n:0 ` +1] `:0 f:95 := homotopy_group_homomorphism n f
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end eq
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