finish first part of constructing gysin sequence

We have a long exact sequence, we still need to show that it consists of the correct groups

algebra/graded.hlean

@@ -673,6 +673,13 @@ begin
   exact graded_hom_square f (to_right_inv (deg f) (deg f x)) idp (to_left_inv (deg f) x) idp
 end
 
+definition graded_homology_isomorphism_kernel_module
+  (g : M₂ →gm M₃) (f : M₁ →gm M₂) (x : I)
+  (H : Πm, image (f ← x) m → m = 0) : graded_homology g f x ≃lm graded_kernel g x :=
+begin
+  apply quotient_module_isomorphism, intro m h, apply subtype_eq, apply H, exact h
+end
+
 definition image_of_graded_homology_intro_eq_zero {g : M₂ →gm M₃} {f : M₁ →gm M₂}
   ⦃i j : I⦄ (p : deg f i = j) (m : graded_kernel g j) (H : graded_homology_intro g f j m = 0) :
   image (f ↘ p) m.1 :=

algebra/left_module.hlean

@@ -312,6 +312,13 @@ section
   definition lm_constant [constructor] (M₁ M₂ : LeftModule R) : M₁ →lm M₂ :=
   homomorphism.mk (const M₁ 0) !is_module_hom_constant
 
+  definition trivial_image_of_is_contr {R} {M₁ M₂ : LeftModule R} {φ : M₁ →lm M₂} (H : is_contr M₁)
+    ⦃m : M₂⦄ (hm : image φ m) : m = 0 :=
+  begin
+    induction hm with m' p, induction p,
+    exact ap φ (@eq_of_is_contr _ H _ _) ⬝ to_respect_zero φ
+  end
+
   structure isomorphism (M₁ M₂ : LeftModule R) :=
     (to_hom : M₁ →lm M₂)
     (is_equiv_to_hom : is_equiv to_hom)

algebra/submodule.hlean

@@ -468,10 +468,16 @@ definition is_surjective_of_is_contr_homology_of_is_contr (ψ : M₂ →lm M₃)
   (H₁ : is_contr (homology ψ φ)) (H₂ : is_contr M₃) : is_surjective φ :=
 is_surjective_of_is_contr_homology_of_constant ψ H₁ (λm, @eq_of_is_contr _ H₂ _ _)
 
+definition homology_isomorphism_kernel_module (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂)
+  (H : Πm, image φ m → m = 0) : homology ψ φ ≃lm kernel_module ψ :=
+begin
+  apply quotient_module_isomorphism, intro m h, apply subtype_eq, apply H, exact h
+end
+
 definition cokernel_module (φ : M₁ →lm M₂) : LeftModule R :=
 quotient_module (image φ)
 
-definition cokernel_module_isomorphism_homology (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) (H : Πm, ψ m = 0) :
+definition homology_isomorphism_cokernel_module (ψ : M₂ →lm M₃) (φ : M₁ →lm M₂) (H : Πm, ψ m = 0) :
   homology ψ φ ≃lm cokernel_module φ :=
 quotient_module_isomorphism_quotient_module
   (submodule_isomorphism _ H)

cohomology/gysin.hlean

@@ -7,22 +7,48 @@ open eq pointed is_trunc is_conn is_equiv equiv sphere fiber chain_complex left_
      prod nat int algebra function spectral_sequence
 
 namespace cohomology
--- set_option pp.universes true
+/-
--- print unreduced_ordinary_cohomology_sphere_of_neq_nat
+  We have maps:
--- --set_option formatter.hide_full_terms false
+  d_m = d_(m-1,n+1)^n : E_(m-1,n+1)^n → E_(m+n+1,0)^n
--- exit
+  Note that ker d_m = E_(m-1,n+1)^∞ and coker d_m = E_(m+n+1,0)^∞.
-definition gysin_sequence' {E B : Type*} (n : ℕ) (HB : is_conn 1 B) (f : E →* B)
+  We have short exact sequences
-  (e : pfiber f ≃* sphere (n+1)) (A : AbGroup) : chain_complex +3ℤ :=
+  coker d_{m-1} → D_{m+n}^∞  → ker d_m
-let c := serre_spectral_sequence_map_of_is_conn pt f (EM_spectrum A) 0 (is_strunc_EM_spectrum A) HB in
+  where D^∞ is the abutment of the spectral sequence.
+  This comes from the spectral sequence, using the fact that coker d_{m-1} and ker d_m are the only
+  two nontrivial groups building up D_{m+n}^∞ (in the filtration of D_{m+n}^∞).
+  We can splice these SESs together to get a LES
+  ... E_(m+n,0)^n → D_{m+n}^∞ → E_(m-1,n+1)^n → E_(m+n+1,0)^n → D_{m+n+1}^∞ ...
+-/
+
+definition gysin_sequence' {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
+  (e : pfiber f ≃* sphere (n+1)) (A : AbGroup) : chain_complex -3ℤ :=
+let c := serre_spectral_sequence_map_of_is_conn pt f (EM_spectrum A) 0
+  (is_strunc_EM_spectrum A) HB in
 let cn : is_normal c := !is_normal_serre_spectral_sequence_map_of_is_conn in
-let deg_d_x : Π(m : ℤ), deg (convergent_spectral_sequence.d c n) ((m+1) - 3, n + 1) = (n + m - 0, 0) :=
+have deg_d_x : Π(m : ℤ), deg (convergent_spectral_sequence.d c n) ((m - 1) - 1, n + 1) =
+  (n + m - 0, 0),
 begin
   intro m, refine deg_d_normal_eq cn _ _ ⬝ _,
   refine prod_eq _ !add.right_inv,
-  refine add.comm4 (m+1) (-3) n 2 ⬝ _,
+  refine ap (λx, x + (n+2)) !sub_sub ⬝ _,
-  exact ap (λx, x - 1) (add.comm (m + 1) n ⬝ (add.assoc n m 1)⁻¹) ⬝ !add.assoc
+  refine add.comm4 m (- 2) n 2 ⬝ _,
-end in
+  refine ap (λx, x + 0) !add.comm
-left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 3, n + 1))
+end,
+have trivial_E : Π(r : ℕ) (p q : ℤ) (hq : q ≠ 0) (hq' : q ≠ of_nat (n+1)),
+  is_contr (convergent_spectral_sequence.E c r (p, q)),
+begin
+  intros, apply is_contr_E, apply is_contr_ordinary_cohomology, esimp,
+  refine is_contr_equiv_closed_rev _ (unreduced_ordinary_cohomology_sphere_of_neq A hq' hq),
+  apply group.equiv_of_isomorphism, apply unreduced_ordinary_cohomology_isomorphism, exact e⁻¹ᵉ*
+end,
+have trivial_E' : Π(r : ℕ) (p q : ℤ) (hq : q > n+1),
+  is_contr (convergent_spectral_sequence.E c r (p, q)),
+begin
+  intros, apply trivial_E r p q,
+  { intro h, subst h, apply not_lt_zero (n+1), exact lt_of_of_nat_lt_of_nat hq },
+  { intro h, subst h, exact lt.irrefl _ hq }
+end,
+left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 1, n + 1))
   begin
     intro m,
     fapply short_exact_mod_isomorphism,
@@ -30,19 +56,19 @@ left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 3, n
     { fapply short_exact_mod_of_is_contr_submodules
         (convergence_0 c (n + m) (λm, neg_zero)),
       { exact zero_lt_succ n },
-      { intro k Hk0 Hkn, apply is_contr_E,
+      { intro k Hk0 Hkn, apply trivial_E, exact λh, Hk0 (of_nat.inj h),
-        apply is_contr_ordinary_cohomology,
+        exact λh, Hkn (of_nat.inj h), }},
-        refine is_contr_equiv_closed_rev _
-                (unreduced_ordinary_cohomology_sphere_of_neq_nat A Hkn Hk0),
-        apply group.equiv_of_isomorphism, apply unreduced_ordinary_cohomology_isomorphism,
-        exact e⁻¹ᵉ* }},
     { symmetry, refine Einf_isomorphism c (n+1) _ _ ⬝lm
-            convergent_spectral_sequence.α c n (n + m - 0, 0) ⬝lm
+        convergent_spectral_sequence.α c n (n + m - 0, 0) ⬝lm
-            isomorphism_of_eq (ap (graded_homology _ _) _) ⬝lm
+        isomorphism_of_eq (ap (graded_homology _ _) _) ⬝lm
-            !graded_homology_isomorphism ⬝lm
+        !graded_homology_isomorphism ⬝lm
-            cokernel_module_isomorphism_homology _ _ _,
+        homology_isomorphism_cokernel_module _ _ _,
-      { exact sorry },
+      { intros r Hr, apply trivial_E', apply of_nat_lt_of_nat_of_lt,
-      { exact sorry },
+        rewrite [zero_add], exact lt_succ_of_le Hr },
+      { intros r Hr, apply is_contr_E, apply is_normal.normal2 cn,
+        refine lt_of_le_of_lt (le_of_eq (ap (λx : ℤ × ℤ, 0 + pr2 x) (is_normal.normal3 cn r))) _,
+        esimp, rewrite [-sub_eq_add_neg], apply sub_lt_of_pos, apply of_nat_lt_of_nat_of_lt,
+        apply succ_pos },
       { exact (deg_d_x m)⁻¹ },
       { intro x, apply @eq_of_is_contr, apply is_contr_E,
         apply is_normal.normal2 cn,
@@ -51,17 +77,32 @@ left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m - 3, n
         refine lt_of_le_of_lt (le_of_eq (zero_add (-(n+1)))) _,
         apply neg_neg_of_pos, apply of_nat_succ_pos }},
     { reflexivity },
-    { exact sorry }
+    { symmetry,
+      refine Einf_isomorphism c (n+1) _ _ ⬝lm
+        convergent_spectral_sequence.α c n (n + m - (n+1), n+1) ⬝lm
+        graded_homology_isomorphism_kernel_module _ _ _ _ ⬝lm
+        isomorphism_of_eq (ap (graded_kernel _) _),
+      { intros r Hr, apply trivial_E', apply of_nat_lt_of_nat_of_lt,
+        apply lt_add_of_pos_right, apply zero_lt_succ },
+      { intros r Hr, apply is_contr_E, apply is_normal.normal2 cn,
+        refine lt_of_le_of_lt (le_of_eq (ap (λx : ℤ × ℤ, (n+1)+pr2 x) (is_normal.normal3 cn r))) _,
+        esimp, rewrite [-sub_eq_add_neg], apply sub_lt_right_of_lt_add,
+        apply of_nat_lt_of_nat_of_lt, rewrite [zero_add], exact lt_succ_of_le Hr },
+      { apply trivial_image_of_is_contr, rewrite [deg_d_inv_eq],
+        apply trivial_E', apply of_nat_lt_of_nat_of_lt,
+        apply lt_add_of_pos_right, apply zero_lt_succ },
+      { refine prod_eq _ rfl, refine ap (add _) !neg_add ⬝ _,
+        refine add.comm4 n m (-n) (- 1) ⬝ _,
+        refine ap (λx, x + _) !add.right_inv ⬝ !zero_add }}
   end
 
---   (λm, short_exact_mod_isomorphism
+-- open fin
---     _
+-- definition gysin_sequence'_zero_equiv {E B : Type*} {n : ℕ} (HB : is_conn 1 B) (f : E →* B)
---     isomorphism.rfl
+--   (e : pfiber f ≃* sphere (n+1)) (A : AbGroup) (m : ℤ) :
---     _
+--   gysin_sequence' HB f e A (m, 0) ≃ _ :=
---     (short_exact_mod_of_is_contr_submodules
+-- _
---       (convergent_HDinf X _)
+
---       (zero_lt_succ n)
+
---       _))
 
 
 end cohomology