prove the Freudenthal Suspension Theorem
This commit is contained in:
Floris van Doorn
parent
a76e4fce08
commit
0483966328
3 changed files with 259 additions and 102 deletions
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@ -1,6 +1,7 @@
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import .LES_of_homotopy_groups homotopy.connectedness homotopy.homotopy_group homotopy.join
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open eq is_trunc pointed homotopy is_equiv fiber equiv trunc nat chain_complex prod fin algebra
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group trunc_index function join pushout
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namespace nat
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open sigma sum
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definition eq_even_or_eq_odd (n : ℕ) : (Σk, 2 * k = n) ⊎ (Σk, 2 * k + 1 = n) :=
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@ -25,61 +26,6 @@ open nat
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namespace is_conn
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theorem is_contr_HG_fiber_of_is_connected {A B : Type*} (k n : ℕ) (f : A →* B)
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[H : is_conn_map 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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-- TODO: use this for trivial_homotopy_group_of_is_conn (in homotopy.homotopy_group)
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theorem is_conn_of_le (A : Type) {n k : ℕ₋₂} (H : n ≤ k) [is_conn k A] : is_conn n A :=
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begin
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apply is_contr_equiv_closed,
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apply trunc_trunc_equiv_left _ n k H
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end
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definition zero_le_of_nat (n : ℕ) : 0 ≤[ℕ₋₂] n :=
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of_nat_le_of_nat (zero_le n)
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local attribute is_conn_map [reducible] --TODO
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theorem is_conn_map_of_le {A B : Type} (f : A → B) {n k : ℕ₋₂} (H : n ≤ k)
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[is_conn_map k f] : is_conn_map n f :=
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λb, is_conn_of_le _ H
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definition is_surjective_trunc_functor {A B : Type} (n : ℕ₋₂) (f : A → B) [H : is_surjective f]
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: is_surjective (trunc_functor n f) :=
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begin
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cases n with n: intro b,
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{ exact tr (fiber.mk !center !is_prop.elim)},
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{ refine @trunc.rec _ _ _ _ _ b, {intro x, exact is_trunc_of_le _ !minus_one_le_succ},
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clear b, intro b, induction H b with v, induction v with a p,
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exact tr (fiber.mk (tr a) (ap tr p))}
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end
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definition is_surjective_cancel_right {A B C : Type} (g : B → C) (f : A → B)
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[H : is_surjective (g ∘ f)] : is_surjective g :=
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begin
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intro c,
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induction H c with v, induction v with a p,
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exact tr (fiber.mk (f a) p)
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end
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-- Lemma 7.5.14
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theorem is_equiv_trunc_functor_of_is_conn_map {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_map n f] : is_equiv (trunc_functor n f) :=
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begin
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fapply adjointify,
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{ intro b, induction b with b, exact trunc_functor n point (center (trunc n (fiber f b)))},
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{ intro b, induction b with b, esimp, generalize center (trunc n (fiber f b)), intro v,
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induction v with v, induction v with a p, esimp, exact ap tr p},
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{ intro a, induction a with a, esimp, rewrite [center_eq (tr (fiber.mk a idp))]}
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end
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theorem trunc_equiv_trunc_of_is_conn_map {A B : Type} (n : ℕ₋₂) (f : A → B)
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[H : is_conn_map n f] : trunc n A ≃ trunc n B :=
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equiv.mk (trunc_functor n f) (is_equiv_trunc_functor_of_is_conn_map n f)
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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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local attribute comm_group.to_group [coercion]
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local attribute is_equiv_tinverse [instance]
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@ -124,20 +70,20 @@ namespace is_conn
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[H : is_conn_map n f] : is_surjective (π→[n + 1] f) :=
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begin
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induction n using rec_on_even_odd with n,
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{ cases n with n,
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{ exact sorry},
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{ have H3 : is_surjective (π→*[2*(succ n) + 1] f ∘* tinverse), from
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{ have H3 : is_surjective (π→*[2*n + 1] f ∘* tinverse), from
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@is_surjective_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (succ n, 2))
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(@is_contr_HG_fiber_of_is_connected A B (2 * succ n) (2 * succ n) f H !le.refl),
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exact @(is_surjective_cancel_right (pmap.to_fun (π→*[2*(succ n) + 1] f)) tinverse) H3}},
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(is_exact_LES_of_homotopy_groups3 f (n, 2))
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(@is_contr_HG_fiber_of_is_connected A B (2 * n) (2 * n) f H !le.refl),
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exact @(is_surjective_cancel_right (pmap.to_fun (π→*[2*n + 1] f)) tinverse) H3},
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{ exact @is_surjective_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (k, 5))
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(@is_contr_HG_fiber_of_is_connected A B (2 * k + 1) (2 * k + 1) f H !le.refl)}
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end
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/- joins -/
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definition join_empty_right [constructor] (A : Type) : join A empty ≃ A :=
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begin
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fapply equiv.MK,
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@ -153,8 +99,6 @@ namespace is_conn
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{ exact empty.elim (pr2 v)}}
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end
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/- joins -/
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definition join_functor [unfold 7] {A A' B B' : Type} (f : A → A') (g : B → B') :
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join A B → join A' B' :=
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begin
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@ -495,7 +495,6 @@ namespace chain_complex
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pt_mul := one_mul,
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mul_pt := mul_one,
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mul_left_inv_pt := mul.left_inv⦄
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end
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-- the following theorems would also be true of the replace "is_contr" by "is_prop"
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@ -1,45 +1,259 @@
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-- TODO: in wedge connectivity and is_conn.elim, unbundle P
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import homotopy.wedge types.pi .LES_applications --TODO: remove
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import homotopy.wedge types.pi
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open eq homotopy is_trunc pointed susp nat pi equiv is_equiv trunc fiber trunc_index
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open eq homotopy is_trunc pointed susp nat pi equiv is_equiv trunc
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section freudenthal
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parameters {A : Type*} (n : ℕ) [is_conn n A]
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--set_option pp.notation false
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protected definition my_wedge_extension.ext : Π {A : Type*} {B : Type*} (n m : ℕ) [cA : is_conn n (carrier A)] [cB : is_conn m (carrier B)]
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(P : carrier A → carrier B → (m+n)-Type) (f : Π (a : carrier A), trunctype.carrier (P a (Point B)))
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(g : Π (b : carrier B), trunctype.carrier (P (Point A) b)),
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f (Point A) = g (Point B) → (Π (a : carrier A) (b : carrier B), trunctype.carrier (P a b)) :=
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sorry
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definition code_fun (a : A) (q : north = north :> susp A)
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: trunc (n * 2) (fiber (pmap.to_fun (loop_susp_unit A)) q) → trunc (n * 2) (fiber merid (q ⬝ merid a)) :=
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definition iterated_loop_ptrunc_pequiv_con (n : ℕ₋₂) (k : ℕ) (A : Type*)
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(p q : Ω[k](ptrunc (n+k) (Ω A))) :
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iterated_loop_ptrunc_pequiv n k (Ω A) (loop_mul trunc_concat p q) =
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trunc_functor2 (loop_mul concat) (iterated_loop_ptrunc_pequiv n k (Ω A) p)
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(iterated_loop_ptrunc_pequiv n k (Ω A) q) :=
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begin
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intro x, induction x with x,
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esimp at *, cases x with a' p,
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-- apply my_wedge_extension.ext n n,
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exact sorry
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revert n p q, induction k with k IH: intro n p q,
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{ reflexivity},
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{ exact sorry}
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end
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definition code (y : susp A) : north = y → Type :=
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susp.rec_on y
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(λp, trunc (2*n) (fiber (loop_susp_unit A) p))
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(λq, trunc (2*n) (fiber merid q))
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theorem elim_type_merid_inv {A : Type} (PN : Type) (PS : Type) (Pm : A → PN ≃ PS)
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(a : A) : transport (susp.elim_type PN PS Pm) (merid a)⁻¹ = to_inv (Pm a) :=
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by rewrite [tr_eq_cast_ap_fn,↑susp.elim_type,ap_inv,elim_merid]; apply cast_ua_inv_fn
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definition is_conn_trunc (A : Type) (n k : ℕ₋₂) [H : is_conn n A]
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: is_conn n (trunc k A) :=
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begin
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intros,
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apply arrow_pathover_constant_right,
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intro q, rewrite [transport_eq_r],
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apply ua,
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exact sorry
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apply is_trunc_equiv_closed, apply trunc_trunc_equiv_trunc_trunc
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end
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definition freudenthal_suspension : is_conn_map (n*2) (loop_susp_unit A) :=
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section open sphere sphere.ops
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definition psphere_succ [unfold_full] (n : ℕ) : S. (n + 1) = psusp (S. n) := idp
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end
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namespace freudenthal section
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parameters {A : Type*} {n : ℕ} [is_conn n A]
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--porting from Agda
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-- definition Q (x : susp A) : Type :=
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-- trunc (k) (north = x)
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definition up (a : A) : north = north :> susp A := -- up a = loop_susp_unit A a
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merid a ⬝ (merid pt)⁻¹
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definition code_merid : A → ptrunc (n + n) A → ptrunc (n + n) A :=
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begin
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have is_conn n (ptrunc (n + n) A), from !is_conn_trunc,
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refine wedge_extension.ext n n (λ x y, ttrunc (n + n) A) _ _ _,
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{ exact tr},
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{ exact id},
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{ reflexivity}
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end
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definition code_merid_β_left (a : A) : code_merid a pt = tr a :=
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by apply wedge_extension.β_left
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definition code_merid_β_right (b : ptrunc (n + n) A) : code_merid pt b = b :=
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by apply wedge_extension.β_right
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definition code_merid_coh : code_merid_β_left pt = code_merid_β_right pt :=
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begin
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symmetry, apply eq_of_inv_con_eq_idp, apply wedge_extension.coh
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end
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definition is_equiv_code_merid (a : A) : is_equiv (code_merid a) :=
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begin
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refine @is_conn.elim (n.-1) _ _ _ _ a,
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{ intro a, apply is_trunc_of_le, apply minus_one_le_succ},
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{ esimp, exact homotopy_closed id (homotopy.symm (code_merid_β_right))}
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end
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definition code_merid_equiv [constructor] (a : A) : trunc (n + n) A ≃ trunc (n + n) A :=
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equiv.mk _ (is_equiv_code_merid a)
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definition code_merid_inv_pt (x : trunc (n + n) A) : (code_merid_equiv pt)⁻¹ x = x :=
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begin
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refine ap010 @(is_equiv.inv _) _ x ⬝ _,
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{ exact homotopy_closed id (homotopy.symm code_merid_β_right)},
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{ apply is_conn.elim_β},
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{ reflexivity}
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end
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definition code [unfold 4] : susp A → Type :=
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susp.elim_type (trunc (n + n) A) (trunc (n + n) A) code_merid_equiv
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definition is_trunc_code (x : susp A) : is_trunc (n + n) (code x) :=
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begin
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induction x with a: esimp,
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{ exact _},
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{ exact _},
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{ apply is_prop.elimo}
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end
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local attribute is_trunc_code [instance]
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definition decode_north [unfold 4] : code north → trunc (n + n) (north = north :> susp A) :=
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trunc_functor (n + n) up
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definition decode_north_pt : decode_north (tr pt) = tr idp :=
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ap tr !con.right_inv
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definition decode_south [unfold 4] : code south → trunc (n + n) (north = south :> susp A) :=
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trunc_functor (n + n) merid
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definition encode' {x : susp A} (p : north = x) : code x :=
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transport code p (tr pt)
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definition encode [unfold 5] {x : susp A} (p : trunc (n + n) (north = x)) : code x :=
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begin
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induction p with p,
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exact transport code p (tr pt)
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end
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theorem encode_decode_north (c : code north) : encode (decode_north c) = c :=
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begin
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have H : Πc, is_trunc (n + n) (encode (decode_north c) = c), from _,
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esimp at *,
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induction c with a,
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rewrite [↑[encode, decode_north, up, code], con_tr, elim_type_merid, ▸*,
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code_merid_β_left, elim_type_merid_inv, ▸*, code_merid_inv_pt]
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end
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definition decode_coh_f (a : A) : tr (up pt) =[merid a] decode_south (code_merid a (tr pt)) :=
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begin
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refine _ ⬝op ap decode_south (code_merid_β_left a)⁻¹,
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apply trunc_pathover,
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apply eq_pathover,
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refine !ap_constant ⬝ph _ ⬝hp !ap_id⁻¹,
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apply square_of_eq,
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exact whisker_right !con.right_inv (merid a)
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end
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definition decode_coh_g (a' : A) : tr (up a') =[merid pt] decode_south (code_merid pt (tr a')) :=
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begin
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refine _ ⬝op ap decode_south (code_merid_β_right (tr a'))⁻¹,
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apply trunc_pathover,
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apply eq_pathover,
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refine !ap_constant ⬝ph _ ⬝hp !ap_id⁻¹,
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apply square_of_eq, refine !inv_con_cancel_right ⬝ !idp_con⁻¹
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end
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definition decode_coh_equality {A : Type} {a a' : A} (p : a = a')
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: whisker_right (con.right_inv p) p = inv_con_cancel_right p p ⬝ (idp_con p)⁻¹ :=
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by induction p; reflexivity
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theorem decode_coh (a : A) : decode_north =[merid a] decode_south :=
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begin
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apply arrow_pathover_left, intro c, esimp at *,
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induction c with a',
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rewrite [↑code, elim_type_merid, ▸*],
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refine wedge_extension.ext n n _ _ _ _ a a',
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{ exact _},
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{ exact decode_coh_f},
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{ exact decode_coh_g},
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{ clear a a', unfold [decode_coh_f, decode_coh_g], refine ap011 concato_eq _ _,
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{ apply ap (λp, trunc_pathover (eq_pathover (_ ⬝ph square_of_eq p ⬝hp _))),
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apply decode_coh_equality},
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{ apply ap (λp, ap decode_south p⁻¹), apply code_merid_coh}}
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end
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definition decode [unfold 4] {x : susp A} (c : code x) : trunc (n + n) (north = x) :=
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begin
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induction x with a,
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{ exact decode_north c},
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{ exact decode_south c},
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{ exact decode_coh a}
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end
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theorem decode_encode {x : susp A} (p : trunc (n + n) (north = x)) : decode (encode p) = p :=
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begin
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induction p with p, induction p, esimp, apply decode_north_pt
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end
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parameters (A n)
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definition equiv' : trunc (n + n) A ≃ trunc (n + n) (Ω (psusp A)) :=
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equiv.MK decode_north encode decode_encode encode_decode_north
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definition pequiv' : ptrunc (n + n) A ≃* ptrunc (n + n) (Ω (psusp A)) :=
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pequiv_of_equiv equiv' decode_north_pt
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-- can we get this?
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-- definition freudenthal_suspension : is_conn_map (n+n) (loop_susp_unit A) :=
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-- begin
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-- intro p, esimp at *, fapply is_contr.mk,
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-- { note c := encode (tr p), esimp at *, induction c with a, },
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-- { exact sorry}
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-- end
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-- {- Used to prove stability in iterated suspensions -}
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-- module FreudenthalIso
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-- {i} (n : ℕ₋₂) (k : ℕ) (t : k ≠ O) (kle : ⟨ k ⟩ ≤T S (n +2+ S n))
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-- (X : Ptd i) (cX : is-connected (S (S n)) (fst X)) where
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-- open FreudenthalEquiv n (⟨ k ⟩) kle (fst X) (snd X) cX public
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-- hom : Ω^-Group k t (⊙Trunc ⟨ k ⟩ X) Trunc-level
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-- →ᴳ Ω^-Group k t (⊙Trunc ⟨ k ⟩ (⊙Ω (⊙Susp X))) Trunc-level
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-- hom = record {
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-- f = fst F;
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-- pres-comp = ap^-conc^ k t (decodeN , decodeN-pt) }
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-- where F = ap^ k (decodeN , decodeN-pt)
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-- iso : Ω^-Group k t (⊙Trunc ⟨ k ⟩ X) Trunc-level
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-- ≃ᴳ Ω^-Group k t (⊙Trunc ⟨ k ⟩ (⊙Ω (⊙Susp X))) Trunc-level
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-- iso = (hom , is-equiv-ap^ k (decodeN , decodeN-pt) (snd eq))
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end end freudenthal
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open algebra
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definition freudenthal_pequiv (A : Type*) {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
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: ptrunc k A ≃* ptrunc k (Ω (psusp A)) :=
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have H' : k ≤[ℕ₋₂] n + n,
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by rewrite [mul.comm at H, -algebra.zero_add n at {1}]; exact of_nat_le_of_nat H,
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ptrunc_pequiv_ptrunc_of_le H' (freudenthal.pequiv' A n)
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definition freudenthal_equiv {A : Type*} {n k : ℕ} [is_conn n A] (H : k ≤ 2 * n)
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: trunc k A ≃ trunc k (Ω (psusp A)) :=
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freudenthal_pequiv A H
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namespace sphere
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open ops algebra pointed function
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definition stability_pequiv (k n : ℕ) (H : k + 2 ≤ 2 * n) : π*[k + 1] (S. (n+1)) ≃* π*[k] (S. n) :=
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begin
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have H' : k ≤ 2 * pred n,
|
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begin
|
||||
rewrite [mul_pred_right], change pred (pred (k + 2)) ≤ pred (pred (2 * n)),
|
||||
apply pred_le_pred, apply pred_le_pred, exact H
|
||||
end,
|
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have is_conn (of_nat (pred n)) (S. n),
|
||||
begin
|
||||
cases n with n,
|
||||
{ exfalso, exact not_succ_le_zero _ H},
|
||||
{ esimp, apply is_conn_psphere}
|
||||
end,
|
||||
refine pequiv_of_eq (ap (ptrunc 0) (loop_space_succ_eq_in (S. (n+1)) k)) ⬝e* _,
|
||||
rewrite psphere_succ,
|
||||
refine !phomotopy_group_pequiv_loop_ptrunc ⬝e* _,
|
||||
refine loopn_pequiv_loopn k (freudenthal_pequiv _ H')⁻¹ᵉ* ⬝e* _,
|
||||
exact !phomotopy_group_pequiv_loop_ptrunc⁻¹ᵉ*,
|
||||
end
|
||||
print phomotopy_group_pequiv_loop_ptrunc
|
||||
print iterated_loop_ptrunc_pequiv
|
||||
definition to_fun_stability_pequiv (k n : ℕ) (H : k + 2 ≤ 2 * n) --(p : π*[k + 1] (S. (n+1)))
|
||||
: stability_pequiv k n H = _ ∘ _ ∘ cast (ap (ptrunc 0) (loop_space_succ_eq_in (S. (n+1)) k)) :=
|
||||
sorry
|
||||
|
||||
-- definition stability (k n : ℕ) (H : k + 3 ≤ 2 * n) : πg[k+1 +1] (S. (n+1)) = πg[k+1] (S. n) :=
|
||||
-- begin
|
||||
-- fapply Group_eq,
|
||||
-- { refine equiv_of_pequiv (stability_pequiv _ _ _), rewrite succ_add, exact H},
|
||||
-- { intro g h, esimp, }
|
||||
-- end
|
||||
|
||||
end sphere
|
||||
|
||||
end freudenthal
|
||||
/-
|
||||
changes in book:
|
||||
proof 8.6.15: also mention that we ignore multiplication
|
||||
proof 8.4.4: respects points
|
||||
proof 8.4.8: do k=0 separately
|
||||
-/
|
||||
|
|
|
|||
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Reference in a new issue