progress on atiyah-hirzebruch and serre spectral sequences
Note: the Serre spectral sequence only works for unreduced cohomology, so we need some results for that For reduced homology we might get a similar result if we replace the sigma in the RHS by a dependent version of the smash product
This commit is contained in:
Floris van Doorn
parent
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commit
63ec1b8d37
8 changed files with 237 additions and 24 deletions
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@ -570,4 +570,11 @@ namespace EM
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abstract (EMadd1_functor_gcompose φ φ⁻¹ᵍ n)⁻¹* ⬝* EMadd1_functor_phomotopy proof right_inv φ qed n ⬝*
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EMadd1_functor_gid H n end
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definition EM_pequiv_EM (n : ℕ) {G H : AbGroup} (φ : G ≃g H) : K G n ≃* K H n :=
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begin
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cases n with n,
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{ exact pequiv_of_isomorphism φ },
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{ exact EMadd1_pequiv_EMadd1 n φ }
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end
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end EM
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@ -20,31 +20,64 @@ namespace cohomology
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definition cohomology (X : Type*) (Y : spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_pmap X (Y (n+2))
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definition parametrized_cohomology {X : Type*} (Y : X → spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_ppi (λx, Y x (n+2))
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definition ordinary_cohomology [reducible] (X : Type*) (G : AbGroup) (n : ℤ) : AbGroup :=
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cohomology X (EM_spectrum G) n
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definition ordinary_cohomology_Z [reducible] (X : Type*) (n : ℤ) : AbGroup :=
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ordinary_cohomology X agℤ n
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notation `H^` n `[`:0 X:0 `, ` Y:0 `]`:0 := cohomology X Y n
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notation `H^` n `[`:0 X:0 `]`:0 := ordinary_cohomology_Z X n
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definition unreduced_cohomology (X : Type) (Y : spectrum) (n : ℤ) : AbGroup :=
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cohomology X₊ Y n
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definition unreduced_ordinary_cohomology [reducible] (X : Type) (G : AbGroup) (n : ℤ) : AbGroup :=
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unreduced_cohomology X (EM_spectrum G) n
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definition unreduced_ordinary_cohomology_Z [reducible] (X : Type) (n : ℤ) : AbGroup :=
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unreduced_ordinary_cohomology X agℤ n
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definition parametrized_cohomology {X : Type*} (Y : X → spectrum) (n : ℤ) : AbGroup :=
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AbGroup_trunc_ppi (λx, Y x (n+2))
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definition ordinary_parametrized_cohomology [reducible] {X : Type*} (G : X → AbGroup) (n : ℤ) :
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AbGroup :=
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parametrized_cohomology (λx, EM_spectrum (G x)) n
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definition unreduced_parametrized_cohomology {X : Type} (Y : X → spectrum) (n : ℤ) : AbGroup :=
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@parametrized_cohomology X₊ (λx, option.cases_on x sunit Y) n
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definition unreduced_ordinary_parametrized_cohomology [reducible] {X : Type} (G : X → AbGroup)
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(n : ℤ) : AbGroup :=
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unreduced_parametrized_cohomology (λx, EM_spectrum (G x)) n
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notation `H^` n `[`:0 X:0 `, ` Y:0 `]`:0 := cohomology X Y n
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notation `oH^` n `[`:0 X:0 `, ` G:0 `]`:0 := ordinary_cohomology X G n
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notation `H^` n `[`:0 X:0 `]`:0 := ordinary_cohomology_Z X n
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notation `uH^` n `[`:0 X:0 `, ` Y:0 `]`:0 := unreduced_cohomology X Y n
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notation `uoH^` n `[`:0 X:0 `, ` G:0 `]`:0 := unreduced_ordinary_cohomology X G n
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notation `uH^` n `[`:0 X:0 `]`:0 := unreduced_ordinary_cohomology_Z X n
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notation `pH^` n `[`:0 binders `, ` r:(scoped Y, parametrized_cohomology Y n) `]`:0 := r
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notation `opH^` n `[`:0 binders `, ` r:(scoped G, ordinary_parametrized_cohomology G n) `]`:0 := r
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notation `upH^` n `[`:0 binders `, ` r:(scoped Y, unreduced_parametrized_cohomology Y n) `]`:0 := r
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notation `uopH^` n `[`:0 binders `, ` r:(scoped G, unreduced_ordinary_parametrized_cohomology G n) `]`:0 := r
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-- check H^3[S¹*,EM_spectrum agℤ]
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-- check H^3[S¹*]
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-- check pH^3[(x : S¹*), EM_spectrum agℤ]
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/- an alternate definition of cohomology -/
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definition cohomology_equiv_shomotopy_group_cotensor (X : Type*) (Y : spectrum) (n : ℤ) :
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definition cohomology_equiv_shomotopy_group_sp_cotensor (X : Type*) (Y : spectrum) (n : ℤ) :
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H^n[X, Y] ≃ πₛ[-n] (sp_cotensor X Y) :=
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trunc_equiv_trunc 0 (!pfunext ⬝e loop_pequiv_loop !pfunext ⬝e loopn_pequiv_loopn 2
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(pequiv_of_eq (ap (λn, ppmap X (Y n)) (add.comm n 2 ⬝ ap (add 2) !neg_neg⁻¹))))
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definition unpointed_cohomology (X : Type) (Y : spectrum) (n : ℤ) : AbGroup :=
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cohomology X₊ Y n
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definition cohomology_isomorphism_shomotopy_group_sp_cotensor (X : Type*) (Y : spectrum) {n m : ℤ}
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(p : -m = n) : H^n[X, Y] ≃g πₛ[m] (sp_cotensor X Y) :=
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sorry
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definition parametrized_cohomology_isomorphism_shomotopy_group_spi {X : Type*} (Y : X → spectrum)
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{n m : ℤ} (p : -m = n) : pH^n[(x : X), Y x] ≃g πₛ[m] (spi X Y) :=
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sorry
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/- functoriality -/
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@ -81,6 +114,23 @@ definition cohomology_isomorphism_refl (X : Type*) (Y : spectrum) (n : ℤ) (x :
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cohomology_isomorphism (pequiv.refl X) Y n x = x :=
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!Group_trunc_pmap_isomorphism_refl
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definition cohomology_isomorphism_right (X : Type*) {Y Y' : spectrum} (e : Πn, Y n ≃* Y' n) (n : ℤ)
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: H^n[X, Y] ≃g H^n[X, Y'] :=
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sorry
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definition parametrized_cohomology_isomorphism_right (X : Type*) {Y Y' : X → spectrum}
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(e : Πx n, Y x n ≃* Y' x n) (n : ℤ)
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: pH^n[(x : X), Y x] ≃g pH^n[(x : X), Y' x] :=
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sorry
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definition ordinary_cohomology_isomorphism_right (X : Type*) {G G' : AbGroup} (e : G ≃g G')
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(n : ℤ) : oH^n[X, G] ≃g oH^n[X, G'] :=
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cohomology_isomorphism_right X (EM_spectrum_pequiv e) n
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definition ordinary_parametrized_cohomology_isomorphism_right (X : Type*) {G G' : X → AbGroup}
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(e : Πx, G x ≃g G' x) (n : ℤ) : opH^n[(x : X), G x] ≃g opH^n[(x : X), G' x] :=
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parametrized_cohomology_isomorphism_right X (λx, EM_spectrum_pequiv (e x)) n
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/- suspension axiom -/
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definition cohomology_psusp_2 (Y : spectrum) (n : ℤ) :
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@ -169,6 +219,11 @@ theorem EM_dimension (G : AbGroup) (n : ℤ) (H : n ≠ 0) :
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(equiv_of_isomorphism (cohomology_isomorphism (pequiv_plift pbool) _ _)))
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(EM_dimension' G n H)
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open group algebra
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theorem ordinary_cohomology_pbool (G : AbGroup) : ordinary_cohomology pbool G 0 ≃g G :=
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sorry
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--isomorphism_of_equiv (trunc_equiv_trunc 0 (ppmap_pbool_pequiv _ ⬝e _) ⬝e !trunc_equiv) sorry
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/- cohomology theory -/
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structure cohomology_theory.{u} : Type.{u+1} :=
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@ -1,9 +1,12 @@
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import ..algebra.module_exact_couple .strunc
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import ..algebra.module_exact_couple .strunc .cohomology
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open eq spectrum trunc is_trunc pointed int EM algebra left_module fiber lift equiv is_equiv
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cohomology group sigma unit
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set_option pp.binder_types true
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/- Eilenberg MacLane spaces are the fibers of the Postnikov system of a type -/
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namespace pointed
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definition postnikov_map [constructor] (A : Type*) (n : ℕ₋₂) : ptrunc (n.+1) A →* ptrunc n A :=
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ptrunc.elim (n.+1) !ptr
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@ -62,6 +65,10 @@ begin
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end,
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this⁻¹ᵛ*
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end pointed open pointed
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namespace spectrum
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definition is_strunc_strunc_pred (X : spectrum) (k : ℤ) : is_strunc k (strunc (k - 1) X) :=
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λn, @(is_trunc_of_le _ (maxm2_monotone (add_le_add_right (sub_one_le k) n))) !is_strunc_strunc
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@ -69,10 +76,11 @@ definition postnikov_smap [constructor] (X : spectrum) (k : ℤ) :
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strunc k X →ₛ strunc (k - 1) X :=
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strunc_elim (str (k - 1) X) (is_strunc_strunc_pred X k)
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definition postnikov_smap_phomotopy [constructor] (X : spectrum) (k : ℤ) (n : ℤ) :
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postnikov_smap X k n ~* postnikov_map (X n) (maxm2 (k - 1 + n)) ∘*
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sorry :=
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sorry
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definition pfiber_postnikov_smap (A : spectrum) (n : ℤ) (k : ℤ) :
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pfiber (postnikov_smap A n k) ≃* EM_spectrum (πₛ[n] A) k :=
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begin
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exact sorry
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end
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section atiyah_hirzebruch
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parameters {X : Type*} (Y : X → spectrum) (s₀ : ℤ) (H : Πx, is_strunc s₀ (Y x))
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@ -80,16 +88,95 @@ section atiyah_hirzebruch
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definition atiyah_hirzebruch_exact_couple : exact_couple rℤ Z2 :=
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@exact_couple_sequence (λs, strunc s (spi X Y)) (postnikov_smap (spi X Y))
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definition atiyah_hirzebruch_ub ⦃s n : ℤ⦄ (Hs : s ≤ n - 1) :
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is_contr (πₛ[n] (strunc s (spi X Y))) :=
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begin
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apply trivial_shomotopy_group_of_is_strunc,
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apply is_strunc_strunc,
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exact lt_of_le_sub_one Hs
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end
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include H
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definition atiyah_hirzebruch_lb ⦃s n : ℤ⦄ (Hs : s ≥ s₀ + 1) :
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is_equiv (postnikov_smap (spi X Y) s n) :=
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begin
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have H2 : is_strunc s₀ (spi X Y), from is_strunc_spi _ _ H,
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refine is_equiv_of_equiv_of_homotopy
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(ptrunc_pequiv_ptrunc_of_is_trunc _ _ (H2 n)) _,
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{ apply maxm2_monotone, apply add_le_add_right, exact le.trans !le_add_one Hs },
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{ apply maxm2_monotone, apply add_le_add_right, exact le_sub_one_of_lt Hs },
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refine @trunc.rec _ _ _ _ _,
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{ intro x, apply is_trunc_eq,
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assert H3 : maxm2 (s - 1 + n) ≤ (maxm2 (s + n)).+1,
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{ refine trunc_index.le_succ (maxm2_monotone (le.trans (le_of_eq !add.right_comm)
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!sub_one_le)) },
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exact @is_trunc_of_le _ _ _ H3 !is_trunc_trunc },
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intro a, reflexivity
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end
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definition is_bounded_atiyah_hirzebruch : is_bounded atiyah_hirzebruch_exact_couple :=
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is_bounded_sequence _ s₀ (λn, n - 1)
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is_bounded_sequence _ s₀ (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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definition atiyah_hirzebruch_convergence' :
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(λn s, πₛ[n] (sfiber (postnikov_smap (spi X Y) s))) ⟹ᵍ (λn, πₛ[n] (strunc s₀ (spi X Y))) :=
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converges_to_sequence _ s₀ (λn, n - 1) atiyah_hirzebruch_lb atiyah_hirzebruch_ub
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lemma spi_EM_spectrum (k s : ℤ) :
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EM_spectrum (πₛ[s] (spi X Y)) k ≃* spi X (λx, EM_spectrum (πₛ[s] (Y x))) k :=
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sorry
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definition atiyah_hirzebruch_convergence :
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(λn s, opH^-n[(x : X), πₛ[s] (Y x)]) ⟹ᵍ (λn, pH^-n[(x : X), Y x]) :=
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converges_to_g_isomorphism atiyah_hirzebruch_convergence'
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begin
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intro s n H,
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exact sorry
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intro n s,
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refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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refine pfiber_postnikov_smap (spi X Y) s k ⬝e* _,
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apply spi_EM_spectrum
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end
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begin
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intro s n H, apply trivial_shomotopy_group_of_is_strunc,
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apply is_strunc_strunc,
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exact lt_of_le_sub_one H,
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intro n,
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refine _ ⬝g (parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp)⁻¹ᵍ,
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apply shomotopy_group_isomorphism_of_pequiv, intro k,
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apply ptrunc_pequiv, exact is_strunc_spi s₀ Y H k,
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end
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end atiyah_hirzebruch
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section serre
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variables {X : Type} (F : X → Type) (Y : spectrum) (s₀ : ℤ) (H : is_strunc s₀ Y)
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open option
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definition add_point_over {X : Type} (F : X → Type) (x : option X) : Type* :=
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(option.cases_on x (lift empty) F)₊
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postfix `₊ₒ`:(max+1) := add_point_over
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/- NOTE: we need unreduced cohomology, maybe also define aityah_hirzebruch for unreduced cohomology -/
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include H
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definition serre_convergence :
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(λn s, opH^-n[(x : X₊), H^-s[F₊ₒ x, Y]]) ⟹ᵍ (λn, H^-n[(Σ(x : X), F x)₊, Y]) :=
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-- (λn s, uopH^-n[(x : X), uH^-s[F x, Y]]) ⟹ᵍ (λn, uH^-n[Σ(x : X), F x, Y]) :=
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proof
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converges_to_g_isomorphism
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(atiyah_hirzebruch_convergence (λx, sp_cotensor (F₊ₒ x) Y) s₀
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(λx, is_strunc_sp_cotensor s₀ (F₊ₒ x) H))
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begin
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intro n s,
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apply ordinary_parametrized_cohomology_isomorphism_right,
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intro x,
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exact (cohomology_isomorphism_shomotopy_group_sp_cotensor _ _ idp)⁻¹ᵍ,
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end
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begin
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intro n, exact sorry
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-- refine parametrized_cohomology_isomorphism_shomotopy_group_spi _ idp ⬝g _,
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-- refine _ ⬝g (cohomology_isomorphism_shomotopy_group_sp_cotensor _ _ idp)⁻¹ᵍ,
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-- apply shomotopy_group_isomorphism_of_pequiv, intro k,
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end
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qed
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end serre
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/- TODO: πₛ[n] (strunc 0 (spi X Y)) ≃g H^n[X, λx, Y x] -/
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end spectrum
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@ -8,7 +8,7 @@ Authors: Michael Shulman, Floris van Doorn, Egbert Rijke, Stefano Piceghello, Yu
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import homotopy.LES_of_homotopy_groups .splice ..colim types.pointed2 .EM ..pointed_pi .smash_adjoint ..algebra.seq_colim .fwedge .pointed_cubes
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open eq nat int susp pointed pmap sigma is_equiv equiv fiber algebra trunc trunc_index pi group
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seq_colim succ_str EM EM.ops function
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seq_colim succ_str EM EM.ops function unit lift is_trunc
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/---------------------
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Basic definitions
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@ -253,6 +253,18 @@ namespace spectrum
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-- Equivalences of prespectra
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------------------------------
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definition spectrum_pequiv_of_nat {E F : spectrum} (e : Π(n : ℕ), E n ≃* F n) (n : ℤ) :
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E n ≃* F n :=
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begin
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have Πn, E (n + 1) ≃* F (n + 1) → E n ≃* F n,
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from λk f, equiv_glue E k ⬝e* loop_pequiv_loop f ⬝e* (equiv_glue F k)⁻¹ᵉ*,
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induction n with n n,
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exact e n,
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induction n with n IH,
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{ exact this -[1+0] (e 0) },
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{ exact this -[1+succ n] IH }
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end
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structure is_sequiv {N : succ_str} {E F : gen_prespectrum N} (f : E →ₛ F) : Type :=
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(to_linv : F →ₛ E)
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(is_retr : to_linv ∘ₛf ~ₛ sid E)
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@ -326,6 +338,13 @@ namespace spectrum
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definition psp_sphere : gen_prespectrum +ℕ :=
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psp_susp bool.pbool
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------------------------------
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-- Contractible spectrum
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------------------------------
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definition sunit.{u} [constructor] : spectrum.{u} :=
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spectrum.MK (λn, plift punit) (λn, pequiv_of_is_contr _ _ _ _)
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/---------------------
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Homotopy groups
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---------------------/
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@ -342,6 +361,14 @@ namespace spectrum
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πₛ[n] E →g πₛ[n] F :=
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π→g[2] (f (2 - n))
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definition shomotopy_group_isomorphism_of_pequiv (n : ℤ) {E F : spectrum} (f : Πn, E n ≃* F n) :
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πₛ[n] E ≃g πₛ[n] F :=
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homotopy_group_isomorphism_of_pequiv 1 (f (2 - n))
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definition shomotopy_group_isomorphism_of_pequiv_nat (n : ℕ) {E F : spectrum}
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(f : Πn, E n ≃* F n) : πₛ[n] E ≃g πₛ[n] F :=
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shomotopy_group_isomorphism_of_pequiv n (spectrum_pequiv_of_nat f)
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notation `πₛ→[`:95 n:0 `]`:0 := shomotopy_group_fun n
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-- what an awful name
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@ -405,7 +432,8 @@ namespace spectrum
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Fibers and long exact sequences
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-----------------------------------------/
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definition sfiber {N : succ_str} {X Y : gen_spectrum N} (f : X →ₛ Y) : gen_spectrum N :=
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definition sfiber [constructor] {N : succ_str} {X Y : gen_spectrum N} (f : X →ₛ Y) :
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gen_spectrum N :=
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spectrum.MK (λn, pfiber (f n))
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(λn, (loop_pfiber (f (S n)))⁻¹ᵉ* ∘*ᵉ pfiber_pequiv_of_square _ _ (sglue_square f n))
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@ -728,6 +756,10 @@ spectrify_fun (smash_prespectrum_fun f g)
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definition EM_spectrum /-[constructor]-/ (G : AbGroup) : spectrum :=
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spectrum.Mk (K G) (λn, (loop_EM G n)⁻¹ᵉ*)
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definition EM_spectrum_pequiv {G H : AbGroup} (e : G ≃g H) (n : ℤ) :
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EM_spectrum G n ≃* EM_spectrum H n :=
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spectrum_pequiv_of_nat (λk, EM_pequiv_EM k e) n
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/- Wedge of prespectra -/
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open fwedge
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|
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@ -134,6 +134,10 @@ begin
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(maxm2_monotone (algebra.add_le_add_right H n))
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end
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definition is_strunc_pequiv_closed {k : ℤ} {E F : spectrum} (H : Πn, E n ≃* F n)
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(H2 : is_strunc k E) : is_strunc k F :=
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λn, is_trunc_equiv_closed (maxm2 (k + n)) (H n)
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definition is_strunc_strunc (k : ℤ) (E : spectrum)
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: is_strunc k (strunc k E) :=
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λ n, is_trunc_trunc (maxm2 (k + n)) (E n)
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@ -255,7 +259,7 @@ section
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|
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end
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definition is_strunc_spi (A : Type*) (k n : ℤ) (H : n ≤ k) (P : A → n-spectrum)
|
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definition is_strunc_spi_of_le {A : Type*} (k n : ℤ) (H : n ≤ k) (P : A → n-spectrum)
|
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: is_strunc k (spi A P) :=
|
||||
begin
|
||||
assert K : n ≤ -[1+ 0] + 1 + k,
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|
|
@ -266,4 +270,12 @@ begin
|
|||
(truncspectrum.struct (P a)))) }
|
||||
end
|
||||
|
||||
definition is_strunc_spi {A : Type*} (n : ℤ) (P : A → spectrum) (H : Πa, is_strunc n (P a))
|
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: is_strunc n (spi A P) :=
|
||||
is_strunc_spi_of_le n n !le.refl (λa, truncspectrum.mk (P a) (H a))
|
||||
|
||||
definition is_strunc_sp_cotensor (n : ℤ) (A : Type*) {Y : spectrum} (H : is_strunc n Y)
|
||||
: is_strunc n (sp_cotensor A Y) :=
|
||||
is_strunc_pequiv_closed (λn, !pppi_pequiv_ppmap) (is_strunc_spi n (λa, Y) (λa, H))
|
||||
|
||||
end spectrum
|
||||
|
|
|
|||
|
|
@ -227,6 +227,12 @@ namespace int
|
|||
definition sub_one_le (n : ℤ) : n - 1 ≤ n :=
|
||||
sub_nat_le n 1
|
||||
|
||||
definition le_add_nat (n : ℤ) (m : ℕ) : n ≤ n + m :=
|
||||
le.intro rfl
|
||||
|
||||
definition le_add_one (n : ℤ) : n ≤ n + 1:=
|
||||
le_add_nat n 1
|
||||
|
||||
end int
|
||||
|
||||
namespace pmap
|
||||
|
|
@ -236,6 +242,14 @@ namespace pmap
|
|||
|
||||
end pmap
|
||||
|
||||
namespace lift
|
||||
|
||||
definition is_trunc_plift [instance] [priority 1450] (A : Type*) (n : ℕ₋₂)
|
||||
[H : is_trunc n A] : is_trunc n (plift A) :=
|
||||
is_trunc_lift A n
|
||||
|
||||
end lift
|
||||
|
||||
namespace trunc
|
||||
|
||||
-- TODO: redefine loopn_ptrunc_pequiv
|
||||
|
|
|
|||
|
|
@ -195,6 +195,9 @@ namespace pointed
|
|||
definition is_contr_loop (A : Type*) [is_set A] : is_contr (Ω A) :=
|
||||
is_contr.mk idp (λa, !is_prop.elim)
|
||||
|
||||
definition is_contr_punit [instance] : is_contr punit :=
|
||||
is_contr_unit
|
||||
|
||||
definition pequiv_of_is_contr (A B : Type*) (HA : is_contr A) (HB : is_contr B) : A ≃* B :=
|
||||
pequiv_punit_of_is_contr A _ ⬝e* (pequiv_punit_of_is_contr B _)⁻¹ᵉ*
|
||||
|
||||
|
|
|
|||
|
|
@ -399,7 +399,7 @@ namespace pointed
|
|||
idpo)⁻¹ᵉ*
|
||||
qed
|
||||
|
||||
definition pmap_psigma {A B : Type*} (C : B → Type) (c : C pt) :
|
||||
definition ppmap_psigma {A B : Type*} (C : B → Type) (c : C pt) :
|
||||
ppmap A (psigma_gen C c) ≃*
|
||||
psigma_gen (λ(f : ppmap A B), ppi_gen (C ∘ f) (transport C (respect_pt f)⁻¹ c))
|
||||
(ppi_const _) :=
|
||||
|
|
@ -430,7 +430,7 @@ namespace pointed
|
|||
refine !idp_con ⬝ _, symmetry, refine !ap_id ◾ !idp_con ⬝ _, apply con.right_inv
|
||||
end
|
||||
... ≃* ppmap A (psigma_gen (λb, f b = pt) (respect_pt f)) :
|
||||
by exact (pmap_psigma _ _)⁻¹ᵉ*
|
||||
by exact (ppmap_psigma _ _)⁻¹ᵉ*
|
||||
... ≃* ppmap A (pfiber f) : by exact pequiv_ppcompose_left !pfiber.sigma_char'⁻¹ᵉ*
|
||||
|
||||
|
||||
|
|
@ -460,6 +460,9 @@ namespace pointed
|
|||
by exact (ppi_psigma _ _)⁻¹ᵉ*
|
||||
... ≃* Π*(a : A), pfiber (f a) : by exact ppi_pequiv_right (λa, !pfiber.sigma_char'⁻¹ᵉ*)
|
||||
|
||||
-- definition pppi_ppmap {A C : Type*} {B : A → Type*} :
|
||||
-- ppmap (/- dependent smash of B -/) C ≃* Π*(a : A), ppmap (B a) C :=
|
||||
|
||||
end pointed open pointed
|
||||
|
||||
open is_trunc is_conn
|
||||
|
|
|
|||
Loading…
Reference in a new issue