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prove the other sorry's in cohomology

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Floris van Doorn 2017-07-08 15:40:43 +01:00
parent e6b1c49f4a
commit 0f24cda263
2 changed files with 17 additions and 8 deletions
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20
homotopy/cohomology.hlean
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@ -90,7 +90,12 @@ end
definition cohomology_isomorphism_shomotopy_group_sp_cotensor (X : Type*) (Y : spectrum) {n m : ℤ}
(p : -m = n) : H^n[X, Y] ≃g πₛ[m] (sp_cotensor X Y) :=
sorry /- TODO FOR SSS -/
begin
refine !trunc_ppi_isomorphic_pmap⁻¹ᵍ ⬝g _,
refine parametrized_cohomology_isomorphism_shomotopy_group_spi (λx, Y) p ⬝g _,
apply shomotopy_group_isomorphism_of_pequiv, intro k,
apply pppi_pequiv_ppmap
end
definition unreduced_cohomology_isomorphism_shomotopy_group_sp_ucotensor (X : Type) (Y : spectrum)
{n m : ℤ} (p : -m = n) : uH^n[X, Y] ≃g πₛ[m] (sp_ucotensor X Y) :=
@ -99,8 +104,6 @@ begin
apply shomotopy_group_isomorphism_of_pequiv, intro k, apply ppmap_add_point
end
/- functoriality -/
definition cohomology_functor [constructor] {X X' : Type*} (f : X' →* X) (Y : spectrum)
@ -138,23 +141,24 @@ definition cohomology_isomorphism_refl (X : Type*) (Y : spectrum) (n : ℤ) (x :
definition cohomology_isomorphism_right (X : Type*) {Y Y' : spectrum} (e : Πn, Y n ≃* Y' n)
(n : ℤ) : H^n[X, Y] ≃g H^n[X, Y'] :=
sorry /- TODO FOR SSS -/
cohomology_isomorphism_shomotopy_group_sp_cotensor X Y !neg_neg ⬝g
shomotopy_group_isomorphism_of_pequiv (-n) (λk, pequiv_ppcompose_left (e k)) ⬝g
(cohomology_isomorphism_shomotopy_group_sp_cotensor X Y' !neg_neg)⁻¹ᵍ
definition parametrized_cohomology_isomorphism_right {X : Type*} {Y Y' : X → spectrum}
(e : Πx n, Y x n ≃* Y' x n) (n : ℤ) : pH^n[(x : X), Y x] ≃g pH^n[(x : X), Y' x] :=
parametrized_cohomology_isomorphism_shomotopy_group_spi Y !neg_neg ⬝g
shomotopy_group_isomorphism_of_pequiv (-n) (λk, ppi_pequiv_right sorry) ⬝g
shomotopy_group_isomorphism_of_pequiv (-n) (λk, ppi_pequiv_right (λx, e x k)) ⬝g
(parametrized_cohomology_isomorphism_shomotopy_group_spi Y' !neg_neg)⁻¹ᵍ
--sorry /- TODO FOR SSS -/
definition unreduced_parametrized_cohomology_isomorphism_right {X : Type} {Y Y' : X → spectrum}
(e : Πx n, Y x n ≃* Y' x n) (n : ℤ) : upH^n[(x : X), Y x] ≃g upH^n[(x : X), Y' x] :=
sorry /- TODO FOR SSS -/
parametrized_cohomology_isomorphism_right (λx' k, add_point_over_pequiv (λx, e x k) x') n
definition unreduced_ordinary_parametrized_cohomology_isomorphism_right {X : Type}
{G G' : X → AbGroup} (e : Πx, G x ≃g G' x) (n : ℤ) :
uopH^n[(x : X), G x] ≃g uopH^n[(x : X), G' x] :=
sorry /- TODO FOR SSS -/
unreduced_parametrized_cohomology_isomorphism_right (λx, EM_spectrum_pequiv (e x)) n
definition ordinary_cohomology_isomorphism_right (X : Type*) {G G' : AbGroup} (e : G ≃g G')
(n : ℤ) : oH^n[X, G] ≃g oH^n[X, G'] :=

5
pointed.hlean
View file

@ -231,6 +231,11 @@ namespace pointed
| (some a) := B a
| none := plift punit
definition add_point_over_pequiv {A : Type} {B B' : A → Type*} (e : Πa, B a ≃* B' a) :
Π(a : A₊), add_point_over B a ≃* add_point_over B' a
| (some a) := e a
| none := pequiv.rfl
definition phomotopy_group_plift_punit.{u} (n : ℕ) [H : is_at_least_two n] :
πag[n] (plift.{0 u} punit) ≃g trivial_ab_group_lift.{u} :=
begin