fix(LES_of_homotopy_groups): make LES of homotopy groups more usable
This commit is contained in:
Floris van Doorn
parent
8a7319244f
commit
66ea4a4725
7 changed files with 134 additions and 101 deletions
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@ -37,26 +37,31 @@ namespace eq
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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 (Ω[succ n] A)
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section
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local attribute inf_group_loopn [instance]
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definition group_homotopy_group [instance] [constructor] [reducible] (n : ℕ) [is_succ n] (A : Type*)
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: group (π[n] A) :=
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trunc_group (Ω[n] A)
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end
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definition group_homotopy_group2 [instance] (k : ℕ) (A : Type*) :
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group (carrier (ptrunctype.to_pType (π[k + 1] A))) :=
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group_homotopy_group k A
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group_homotopy_group (k+1) A
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definition ab_group_homotopy_group [constructor] [reducible] (n : ℕ) (A : Type*)
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: ab_group (π[succ (succ n)] A) :=
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trunc_ab_group (Ω[succ (succ n)] A)
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section
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local attribute ab_inf_group_loopn [instance]
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definition ab_group_homotopy_group [constructor] [reducible] (n : ℕ) [is_at_least_two n] (A : Type*)
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: ab_group (π[n] A) :=
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trunc_ab_group (Ω[n] A)
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end
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local attribute ab_group_homotopy_group [instance]
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definition ghomotopy_group [constructor] : Π(n : ℕ) [is_succ n] (A : Type*), Group
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| (succ n) x A := Group.mk (π[succ n] A) _
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definition ghomotopy_group [constructor] (n : ℕ) [is_succ n] (A : Type*) : Group :=
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Group.mk (π[n] A) _
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definition cghomotopy_group [constructor] :
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Π(n : ℕ) [is_at_least_two n] (A : Type*), AbGroup
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| (succ (succ n)) x A := AbGroup.mk (π[succ (succ n)] A) _
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definition cghomotopy_group [constructor] (n : ℕ) [is_at_least_two n] (A : Type*) : AbGroup :=
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AbGroup.mk (π[n] A) _
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definition fundamental_group [constructor] (A : Type*) : Group :=
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ghomotopy_group 1 A
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@ -258,12 +263,12 @@ namespace eq
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inv_preserve_binary (homotopy_group_pequiv_loop_ptrunc (succ k) A) mul concat
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(@homotopy_group_pequiv_loop_ptrunc_con k A) p q
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definition ghomotopy_group_ptrunc [constructor] (k : ℕ) (A : Type*) :
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πg[k+1] (ptrunc (k+1) A) ≃g πg[k+1] A :=
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definition ghomotopy_group_ptrunc_of_le [constructor] {k n : ℕ} (H : k ≤ n) [Hk : is_succ k] (A : Type*) :
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πg[k] (ptrunc n A) ≃g πg[k] A :=
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begin
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fapply isomorphism_of_equiv,
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{ exact homotopy_group_ptrunc (k+1) A},
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{ intro g₁ g₂,
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{ exact homotopy_group_ptrunc_of_le H A},
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{ intro g₁ g₂, induction Hk with k,
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refine _ ⬝ !homotopy_group_pequiv_loop_ptrunc_inv_con,
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apply ap ((homotopy_group_pequiv_loop_ptrunc (k+1) A)⁻¹ᵉ*),
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refine _ ⬝ !loopn_pequiv_loopn_con ,
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@ -271,6 +276,10 @@ namespace eq
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apply homotopy_group_pequiv_loop_ptrunc_con}
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end
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definition ghomotopy_group_ptrunc [constructor] (k : ℕ) [is_succ k] (A : Type*) :
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πg[k] (ptrunc k A) ≃g πg[k] A :=
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ghomotopy_group_ptrunc_of_le (le.refl k) A
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/- some homomorphisms -/
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-- definition is_homomorphism_cast_loopn_succ_eq_in {A : Type*} (n : ℕ) :
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@ -75,12 +75,14 @@ We get the long exact sequence of homotopy groups by taking the set-truncation o
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import .chain_complex algebra.homotopy_group eq2
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open eq pointed sigma fiber equiv is_equiv sigma.ops is_trunc nat trunc algebra function sum
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open eq pointed sigma fiber equiv is_equiv is_trunc nat trunc algebra function sum
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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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section
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open sigma.ops
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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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@ -90,7 +92,10 @@ namespace chain_complex
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: Σ(Z X : Type*), Z →* X :=
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iterate fiber_sequence_helper n v
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end
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section
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open sigma.ops
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universe variable u
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parameters {X Y : pType.{u}} (f : X →* Y)
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include f
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@ -470,10 +475,14 @@ namespace chain_complex
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PART 3
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--------------/
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definition fibration_sequence [unfold 4] : fin 3 → Type*
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| (fin.mk 0 H) := Y
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| (fin.mk 1 H) := X
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| (fin.mk 2 H) := pfiber f
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| (fin.mk (n+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition loop_spaces2 [reducible] : +3ℕ → Type*
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| (n, fin.mk 0 H) := Ω[n] Y
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| (n, fin.mk 1 H) := Ω[n] X
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| (n, fin.mk k H) := Ω[n] (pfiber f)
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| (n, m) := Ω[n] (fibration_sequence m)
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definition loop_spaces2_add1 (n : ℕ) : Π(x : fin 3),
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loop_spaces2 (n+1, x) = Ω (loop_spaces2 (n, x))
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@ -629,11 +638,10 @@ namespace chain_complex
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/--------------
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PART 4
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--------------/
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open prod.ops
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definition homotopy_groups [reducible] : +3ℕ → Set*
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| (n, fin.mk 0 H) := π[n] Y
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| (n, fin.mk 1 H) := π[n] X
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| (n, fin.mk k H) := π[n] (pfiber f)
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definition homotopy_groups [reducible] : +3ℕ → Set* :=
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λnm, π[nm.1] (fibration_sequence nm.2)
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definition homotopy_groups_pequiv_loop_spaces2 [reducible]
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: Π(n : +3ℕ), ptrunc 0 (loop_spaces2 n) ≃* homotopy_groups n
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@ -709,27 +717,23 @@ namespace chain_complex
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open group
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definition group_LES_of_homotopy_groups (n : ℕ) : Π(x : fin (succ 2)),
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group (LES_of_homotopy_groups (n + 1, x))
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| (fin.mk 0 H) := begin rexact group_homotopy_group n Y end
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| (fin.mk 1 H) := begin rexact group_homotopy_group n X end
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| (fin.mk 2 H) := begin rexact group_homotopy_group n (pfiber f) end
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| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition group_LES_of_homotopy_groups (n : ℕ) [is_succ n] (x : fin (succ 2)) :
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group (LES_of_homotopy_groups (n, x)) :=
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group_homotopy_group n (fibration_sequence x)
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definition ab_group_LES_of_homotopy_groups (n : ℕ) : Π(x : fin (succ 2)),
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ab_group (LES_of_homotopy_groups (n + 2, x))
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| (fin.mk 0 H) := proof ab_group_homotopy_group n Y qed
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| (fin.mk 1 H) := proof ab_group_homotopy_group n X qed
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| (fin.mk 2 H) := proof ab_group_homotopy_group n (pfiber f) qed
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| (fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition pgroup_LES_of_homotopy_groups (n : ℕ) [H : is_succ n] (x : fin (succ 2)) :
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pgroup (LES_of_homotopy_groups (n, x)) :=
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by induction H with n; exact @pgroup_of_group _ (group_LES_of_homotopy_groups (n+1) x) idp
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definition Group_LES_of_homotopy_groups (x : +3ℕ) : Group.{u} :=
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Group.mk (LES_of_homotopy_groups (nat.succ (pr1 x), pr2 x))
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(group_LES_of_homotopy_groups (pr1 x) (pr2 x))
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definition ab_group_LES_of_homotopy_groups (n : ℕ) [is_at_least_two n] (x : fin (succ 2)) :
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ab_group (LES_of_homotopy_groups (n, x)) :=
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ab_group_homotopy_group n (fibration_sequence x)
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definition Group_LES_of_homotopy_groups (n : +3ℕ) : Group.{u} :=
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πg[n.1+1] (fibration_sequence n.2)
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definition AbGroup_LES_of_homotopy_groups (n : +3ℕ) : AbGroup.{u} :=
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AbGroup.mk (LES_of_homotopy_groups (pr1 n + 2, pr2 n))
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(ab_group_LES_of_homotopy_groups (pr1 n) (pr2 n))
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πag[n.1+2] (fibration_sequence n.2)
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definition homomorphism_LES_of_homotopy_groups_fun : Π(k : +3ℕ),
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Group_LES_of_homotopy_groups (S k) →g Group_LES_of_homotopy_groups k
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@ -748,6 +752,52 @@ namespace chain_complex
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end
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| (k, fin.mk (l+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition LES_is_equiv_of_trivial (n : ℕ) (x : fin (succ 2)) [H : is_succ n]
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(HX1 : is_contr (LES_of_homotopy_groups (stratified_pred snat' (n, x))))
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(HX2 : is_contr (LES_of_homotopy_groups (stratified_pred snat' (n+1, x))))
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: is_equiv (cc_to_fn LES_of_homotopy_groups (n, x)) :=
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begin
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induction H with n,
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induction x with m H, cases m with m,
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{ rexact @is_equiv_of_trivial +3ℕ LES_of_homotopy_groups (n, 2) (is_exact_LES_of_homotopy_groups (n, 2))
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proof (is_exact_LES_of_homotopy_groups (n+1, 0)) qed HX1 proof HX2 qed
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proof pgroup_LES_of_homotopy_groups (n+1) 0 qed proof pgroup_LES_of_homotopy_groups (n+1) 1 qed
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proof homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun (n, 0)) qed },
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cases m with m,
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{ rexact @is_equiv_of_trivial +3ℕ LES_of_homotopy_groups (n+1, 0) (is_exact_LES_of_homotopy_groups (n+1, 0))
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proof (is_exact_LES_of_homotopy_groups (n+1, 1)) qed HX1 proof HX2 qed
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proof pgroup_LES_of_homotopy_groups (n+1) 1 qed proof pgroup_LES_of_homotopy_groups (n+1) 2 qed
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proof homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun (n, 1)) qed }, cases m with m,
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{ rexact @is_equiv_of_trivial +3ℕ LES_of_homotopy_groups (n+1, 1) (is_exact_LES_of_homotopy_groups (n+1, 1))
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proof (is_exact_LES_of_homotopy_groups (n+1, 2)) qed HX1 proof HX2 qed
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proof pgroup_LES_of_homotopy_groups (n+1) 2 qed proof pgroup_LES_of_homotopy_groups (n+2) 0 qed
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proof homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun (n, 2)) qed },
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exfalso, apply lt_le_antisymm H, apply le_add_left
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end
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definition LES_isomorphism_of_trivial_cod (n : ℕ) [H : is_succ n]
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(HX1 : is_contr (πg[n] Y)) (HX2 : is_contr (πg[n+1] Y)) : πg[n] (pfiber f) ≃g πg[n] X :=
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begin
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induction H with n,
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refine isomorphism.mk (homomorphism_LES_of_homotopy_groups_fun (n, 1)) _,
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apply LES_is_equiv_of_trivial, apply HX1, apply HX2
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end
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definition LES_isomorphism_of_trivial_dom (n : ℕ) [H : is_succ n]
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(HX1 : is_contr (πg[n] X)) (HX2 : is_contr (πg[n+1] X)) : πg[n+1] (Y) ≃g πg[n] (pfiber f) :=
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begin
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induction H with n,
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refine isomorphism.mk (homomorphism_LES_of_homotopy_groups_fun (n, 2)) _,
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apply LES_is_equiv_of_trivial, apply HX1, apply HX2
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end
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definition LES_isomorphism_of_trivial_pfiber (n : ℕ)
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(HX1 : is_contr (π[n] (pfiber f))) (HX2 : is_contr (πg[n+1] (pfiber f))) : πg[n+1] X ≃g πg[n+1] Y :=
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begin
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refine isomorphism.mk (homomorphism_LES_of_homotopy_groups_fun (n, 0)) _,
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apply LES_is_equiv_of_trivial, apply HX1, apply HX2
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end
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end
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/-
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@ -794,22 +844,22 @@ namespace chain_complex
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refine _ ⬝* !apn_pcompose⁻¹*, reflexivity end
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| (n, fin.mk (k+3) H) := begin exfalso, apply lt_le_antisymm H, apply le_add_left end
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definition type_fibration_sequence [constructor] : type_chain_complex +3ℕ :=
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definition type_LES_fibration_sequence [constructor] : type_chain_complex +3ℕ :=
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transfer_type_chain_complex
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(LES_of_loop_spaces2 f)
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fibration_sequence_fun
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fibration_sequence_pequiv
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fibration_sequence_fun_phomotopy
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definition is_exact_type_fibration_sequence : is_exact_t type_fibration_sequence :=
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definition is_exact_type_fibration_sequence : is_exact_t type_LES_fibration_sequence :=
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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_LES_of_loop_spaces2
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end
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definition fibration_sequence [constructor] : chain_complex +3ℕ :=
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trunc_chain_complex type_fibration_sequence
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definition LES_fibration_sequence [constructor] : chain_complex +3ℕ :=
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trunc_chain_complex type_LES_fibration_sequence
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end
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@ -49,6 +49,11 @@ definition stratified_succ {N : succ_str} {n : ℕ} (x : stratified_type N n)
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: stratified_type N n :=
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(if val (pr2 x) = n then S (pr1 x) else pr1 x, cyclic_succ (pr2 x))
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/- You might need to manually change the succ_str, to use predecessor as "successor" -/
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definition stratified_pred (N : succ_str) {n : ℕ} (x : stratified_type N n)
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: stratified_type N n :=
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(if val (pr2 x) = 0 then S (pr1 x) else pr1 x, cyclic_pred (pr2 x))
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definition stratified [reducible] [constructor] (N : succ_str) (n : ℕ) : succ_str :=
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succ_str.mk (stratified_type N n) stratified_succ
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@ -7,7 +7,7 @@ Authors: Floris van Doorn, Clive Newstead
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import .LES_of_homotopy_groups .sphere .complex_hopf
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open eq is_trunc trunc_index pointed algebra trunc nat is_conn fiber pointed unit
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open eq is_trunc trunc_index pointed algebra trunc nat is_conn fiber pointed unit group
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namespace is_trunc
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@ -53,15 +53,6 @@ namespace is_trunc
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[H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
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@(trivial_homotopy_group_of_is_conn (pfiber f) H2) (H pt)
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theorem homotopy_group_trunc_of_le (A : Type*) (n k : ℕ) (H : k ≤ n)
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: π[k] (ptrunc n A) ≃* π[k] A :=
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begin
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refine !homotopy_group_pequiv_loop_ptrunc ⬝e* _,
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refine loopn_pequiv_loopn _ (ptrunc_ptrunc_pequiv_left _ _) ⬝e* _,
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exact of_nat_le_of_nat H,
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exact !homotopy_group_pequiv_loop_ptrunc⁻¹ᵉ*,
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end
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/- Corollaries of the LES of homotopy groups -/
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local attribute ab_group.to_group [coercion]
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local attribute is_equiv_tinverse [instance]
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@ -81,16 +72,9 @@ namespace is_trunc
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refine is_conn_fun_of_le f (zero_le_of_nat n)},
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{ /- k > 0 -/
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have H2' : k ≤ n, from le.trans !self_le_succ H2,
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exact
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@is_equiv_of_trivial _
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(LES_of_homotopy_groups f) _
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(is_exact_LES_of_homotopy_groups f (k, 2))
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(is_exact_LES_of_homotopy_groups f (succ k, 0))
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exact LES_is_equiv_of_trivial f (succ k) 0
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(@is_contr_HG_fiber_of_is_connected A B k n f H H2')
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(@is_contr_HG_fiber_of_is_connected A B (succ k) n f H H2)
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(@pgroup_of_group _ (group_LES_of_homotopy_groups f k 0) idp)
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(@pgroup_of_group _ (group_LES_of_homotopy_groups f k 1) idp)
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(homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun f (k, 0)))},
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(@is_contr_HG_fiber_of_is_connected A B (succ k) n f H H2) },
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end
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theorem is_equiv_π_of_is_connected.{u v} {A : pType.{u}} {B : pType.{v}} {n k : ℕ} (f : A →* B)
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@ -131,7 +115,7 @@ namespace is_trunc
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(H : Πa k, is_equiv (π→[k + 1] (pmap_of_map f a))) : is_equiv f :=
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begin
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revert A B HA HB f H' H, induction n with n IH: intros,
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{ apply is_equiv_of_is_contr},
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{ apply is_equiv_of_is_contr },
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have Πa, is_equiv (Ω→ (pmap_of_map f a)),
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begin
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intro a,
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@ -223,8 +207,6 @@ namespace is_trunc
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cases A with A a, exact H k H'
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end
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definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] :
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ab_group (π[n] A) :=
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begin
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@ -233,7 +215,7 @@ namespace is_trunc
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{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
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cases n with n,
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{ unfold [homotopy_group, ptrunc], apply ab_group_of_is_contr },
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exact ab_group_homotopy_group n A
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exact ab_group_homotopy_group (n+2) A
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end
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definition is_contr_of_trivial_homotopy' (n : ℕ₋₂) (A : Type) [is_trunc n A] [is_conn -1 A]
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@ -253,7 +235,7 @@ namespace is_trunc
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intro k a H2,
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induction a with a,
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apply is_trunc_equiv_closed_rev,
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exact equiv_of_pequiv (homotopy_group_trunc_of_le (pointed.MK A a) _ _ H2),
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exact equiv_of_pequiv (homotopy_group_ptrunc_of_le H2 (pointed.MK A a)),
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exact H k a H2
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end
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@ -266,9 +248,6 @@ namespace is_trunc
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cases A with A a, exact H k H2
|
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end
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definition is_conn_fun_of_equiv_on_homotopy_groups.{u} (n : ℕ) {A B : Type.{u}} (f : A → B)
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[is_equiv (trunc_functor 0 f)]
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(H1 : Πa k, k ≤ n → is_equiv (homotopy_group_functor k (pmap_of_map f a)))
|
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|
|
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@ -27,22 +27,12 @@ namespace sphere
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fapply isomorphism_of_equiv,
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{ fapply equiv.mk,
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{ exact cc_to_fn (LES_of_homotopy_groups complex_phopf) (1, 2)},
|
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{ refine @is_equiv_of_trivial _
|
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_ _
|
||||
(is_exact_LES_of_homotopy_groups _ (1, 1))
|
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(is_exact_LES_of_homotopy_groups _ (1, 2))
|
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_
|
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_
|
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(@pgroup_of_group _ (group_LES_of_homotopy_groups complex_phopf _ _) idp)
|
||||
(@pgroup_of_group _ (group_LES_of_homotopy_groups complex_phopf _ _) idp)
|
||||
_,
|
||||
{ rewrite [LES_of_homotopy_groups_1, ▸*],
|
||||
have H : 1 ≤[ℕ] 2, from !one_le_succ,
|
||||
apply trivial_homotopy_group_of_is_conn, exact H, rexact is_conn_psphere 3},
|
||||
{ refine LES_is_equiv_of_trivial complex_phopf 1 2 _ _,
|
||||
{ have H : 1 ≤[ℕ] 2, from !one_le_succ,
|
||||
apply trivial_homotopy_group_of_is_conn, exact H, rexact is_conn_psphere 3 },
|
||||
{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
|
||||
(LES_of_homotopy_groups_1 complex_phopf 2) _,
|
||||
apply trivial_homotopy_group_of_is_conn, apply le.refl, rexact is_conn_psphere 3},
|
||||
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (0, 2))}}},
|
||||
apply trivial_homotopy_group_of_is_conn, apply le.refl, rexact is_conn_psphere 3 }}},
|
||||
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (0, 2))}
|
||||
end
|
||||
|
||||
|
|
@ -54,23 +44,13 @@ namespace sphere
|
|||
{ exact cc_to_fn (LES_of_homotopy_groups complex_phopf) (n+3, 0)},
|
||||
{ have H : is_trunc 1 (pfiber complex_phopf),
|
||||
from @(is_trunc_equiv_closed_rev _ pfiber_complex_phopf) is_trunc_circle,
|
||||
refine @is_equiv_of_trivial _
|
||||
_ _
|
||||
(is_exact_LES_of_homotopy_groups _ (n+2, 2))
|
||||
(is_exact_LES_of_homotopy_groups _ (n+3, 0))
|
||||
_
|
||||
_
|
||||
(@pgroup_of_group _ (group_LES_of_homotopy_groups complex_phopf _ _) idp)
|
||||
(@pgroup_of_group _ (group_LES_of_homotopy_groups complex_phopf _ _) idp)
|
||||
_,
|
||||
{ rewrite [▸*, LES_of_homotopy_groups_2 _ (n +[ℕ] 2)],
|
||||
have H2 : 1 ≤[ℕ] n + 1, from !one_le_succ,
|
||||
exact @trivial_ghomotopy_group_of_is_trunc _ _ _ H H2},
|
||||
refine LES_is_equiv_of_trivial complex_phopf (n+3) 0 _ _,
|
||||
{ have H2 : 1 ≤[ℕ] n + 1, from !one_le_succ,
|
||||
exact @trivial_ghomotopy_group_of_is_trunc _ _ _ H H2 },
|
||||
{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
|
||||
(LES_of_homotopy_groups_2 complex_phopf _) _,
|
||||
have H2 : 1 ≤[ℕ] n + 2, from !one_le_succ,
|
||||
apply trivial_ghomotopy_group_of_is_trunc _ _ _ H2},
|
||||
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (n+2, 0))}}},
|
||||
apply trivial_ghomotopy_group_of_is_trunc _ _ _ H2 }}},
|
||||
{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (n+2, 0))}
|
||||
end
|
||||
|
||||
|
|
|
|||
|
|
@ -595,6 +595,15 @@ end
|
|||
(succ_lt_succ (lt_of_le_of_ne (le_of_lt_succ (is_lt x)) H))}
|
||||
end
|
||||
|
||||
definition cyclic_pred {n : ℕ} (x : fin n) : fin n :=
|
||||
begin
|
||||
cases n with n,
|
||||
{ exfalso, apply not_lt_zero _ (is_lt x)},
|
||||
{ cases x with m H, cases m with m,
|
||||
{ exact fin.mk n (self_lt_succ n) },
|
||||
{ exact fin.mk m (lt.trans (self_lt_succ m) H) }}
|
||||
end
|
||||
|
||||
/-
|
||||
We want to say that fin (succ n) always has a 0 and 1. However, we want a bit more, because
|
||||
sometimes we want a zero of (fin a) where a is either
|
||||
|
|
|
|||
|
|
@ -791,6 +791,7 @@ namespace pointed
|
|||
infix ` ⬝e*p `:75 := peconcat_eq
|
||||
infix ` ⬝pe* `:75 := eq_peconcat
|
||||
|
||||
-- rename pequiv_of_eq_natural
|
||||
definition pequiv_of_eq_commute [constructor] {A : Type} {B C : A → Type*} (f : Πa, B a →* C a)
|
||||
{a₁ a₂ : A} (p : a₁ = a₂) : pequiv_of_eq (ap C p) ∘* f a₁ ~* f a₂ ∘* pequiv_of_eq (ap B p) :=
|
||||
pcast_commute f p
|
||||
|
|
|
|||
Loading…
Reference in a new issue