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compute fibers of postnikov tower

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Floris van Doorn 2017-06-14 20:04:35 -04:00
parent da95ea0acb
commit b6fa4e8716
1 changed files with 32 additions and 44 deletions
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76
homotopy/EM.hlean
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@ -565,58 +565,42 @@ namespace EM
open fiber EM.ops open fiber EM.ops
-- definition loopn_succ_pfiber_postnikov_map (A : Type*) (k : ℕ) (n : ℕ₋₂) :
-- Ω[k+1] (pfiber (postnikov_map A (n.+1))) ≃* Ω[k] (pfiber (postnikov_map (Ω A) n)) :=
-- begin
-- exact sorry
-- end
-- definition loopn_pfiber_postnikov_map (A : Type*) (n : ℕ) :
-- Ω[n] (pfiber (postnikov_map A n)) ≃* EM1 (πg[n+1] A) :=
-- begin
-- revert A, induction n with n IH: intro A,
-- { apply pfiber_postnikov_map_zero },
-- exact loopn_succ_pfiber_postnikov_map A n n ⬝e* IH (Ω A) ⬝e*
-- EM1_pequiv_EM1 !ghomotopy_group_succ_in⁻¹ᵍ
-- end
-- move -- move
definition pgroup_of_Group (X : Group) : pgroup X := definition pgroup_of_Group (X : Group) : pgroup X :=
pgroup_of_group _ idp pgroup_of_group _ idp
open prod chain_complex succ_str fin -- open prod chain_complex succ_str fin
definition isomorphism_of_trivial_LES {A B : Type*} (f : A →* B) (n : ℕ) -- definition isomorphism_of_trivial_LES {A B : Type*} (f : A →* B) (n : ℕ)
(k : fin (nat.succ 2)) (HX1 : is_contr (homotopy_groups f (n+1, k))) -- (k : fin (nat.succ 2)) (HX1 : is_contr (homotopy_groups f (n+1, k)))
(HX2 : is_contr (homotopy_groups f (n+2, k))) : -- (HX2 : is_contr (homotopy_groups f (n+2, k))) :
Group_LES_of_homotopy_groups f (@S +3ℕ (S (n, k))) ≃g Group_LES_of_homotopy_groups f (S (n, k)) := -- Group_LES_of_homotopy_groups f (@S +3ℕ (S (n, k))) ≃g Group_LES_of_homotopy_groups f (S (n, k)) :=
begin -- begin
induction k with k Hk, -- induction k with k Hk,
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 1,
exfalso, apply lt_le_antisymm Hk, apply le_add_left, -- exfalso, apply lt_le_antisymm Hk, apply le_add_left,
all_goals exact let k := fin.mk _ Hk in let x : +3ℕ := (n, k) in let S : +3ℕ → +3ℕ := succ_str.S in -- all_goals exact let k := fin.mk _ Hk in let x : +3ℕ := (n, k) in let S : +3ℕ → +3ℕ := succ_str.S in
let z := -- let z :=
@is_equiv_of_trivial _ -- @is_equiv_of_trivial _
(LES_of_homotopy_groups f) _ -- (LES_of_homotopy_groups f) _
(is_exact_LES_of_homotopy_groups f (n+1, k)) -- (is_exact_LES_of_homotopy_groups f (n+1, k))
(is_exact_LES_of_homotopy_groups f (S (n+1, k))) -- (is_exact_LES_of_homotopy_groups f (S (n+1, k)))
HX1 HX2 -- HX1 HX2
(pgroup_of_Group (Group_LES_of_homotopy_groups f (S x))) -- (pgroup_of_Group (Group_LES_of_homotopy_groups f (S x)))
(pgroup_of_Group (Group_LES_of_homotopy_groups f (S (S x)))) -- (pgroup_of_Group (Group_LES_of_homotopy_groups f (S (S x))))
(homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun f (S x))) in -- (homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun f (S x))) in
isomorphism.mk (homomorphism_LES_of_homotopy_groups_fun f _) z -- isomorphism.mk (homomorphism_LES_of_homotopy_groups_fun f _) z
end -- end
definition pfiber_postnikov_map_zero (A : Type*) : definition pfiber_postnikov_map_zero (A : Type*) :
pfiber (postnikov_map A 0) ≃* EM1 (πg[1] A) := pfiber (postnikov_map A 0) ≃* EM1 (πg[1] A) :=
begin begin
symmetry, apply EM1_pequiv, symmetry, apply EM1_pequiv,
{ symmetry, note z := isomorphism_of_trivial_LES (postnikov_map A 0) 1 0 { symmetry, refine _ ⬝g ghomotopy_group_ptrunc 1 A,
(trivial_homotopy_group_of_is_trunc (ptrunc 0 A) !zero_lt_succ) exact chain_complex.LES_isomorphism_of_trivial_cod _ _
(trivial_homotopy_group_of_is_trunc (ptrunc 0 A) !zero_lt_succ), exact sorry (trivial_homotopy_group_of_is_trunc _ !zero_lt_one)
-- rexact isomorphism_of_equiv (equiv_of_isomorphism z) sorry (trivial_homotopy_group_of_is_trunc _ (zero_lt_succ 1)) },
},
{ apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr }, { apply @is_conn_fun_trunc_elim, apply is_conn_fun_tr },
{ apply is_trunc_pfiber } { apply is_trunc_pfiber }
end end
@ -624,9 +608,13 @@ namespace EM
definition pfiber_postnikov_map_succ (A : Type*) (n : ℕ) : definition pfiber_postnikov_map_succ (A : Type*) (n : ℕ) :
pfiber (postnikov_map A (n+1)) ≃* EMadd1 (πag[n+2] A) (n+1) := pfiber (postnikov_map A (n+1)) ≃* EMadd1 (πag[n+2] A) (n+1) :=
begin begin
apply pequiv_EMadd1_of_loopn_pequiv_EM1, symmetry, apply EMadd1_pequiv,
{ exact sorry }, { refine _ ⬝g ghomotopy_group_ptrunc (n+2) A,
{ apply is_conn_fun_trunc_elim, apply is_conn_fun_tr } exact chain_complex.LES_isomorphism_of_trivial_cod _ _
(trivial_homotopy_group_of_is_trunc _ (self_lt_succ (n+1)))
(trivial_homotopy_group_of_is_trunc _ (le_succ _)) },
{ apply is_conn_fun_trunc_elim, apply is_conn_fun_tr },
{ apply is_trunc_pfiber }
end end