start on gysin sequence

cohomology/gysin.hlean

@@ -0,0 +1,39 @@
+/- the construction of the Gysin sequence using the Serre spectral sequence -/
+-- author: Floris van Doorn
+
+import .serre
+
+open eq pointed is_trunc is_conn is_equiv equiv sphere fiber chain_complex left_module spectrum nat
+     prod nat int algebra
+
+namespace cohomology
+
+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
+left_module.LES_of_SESs _ _ _ (λm, convergent_spectral_sequence.d c n (m, n))
+  begin
+    intro m,
+    fapply short_exact_mod_isomorphism,
+    rotate 3,
+    { fapply short_exact_mod_of_is_contr_submodules
+        (spectral_sequence.convergence_0 c (n + m) (λm, neg_zero)),
+      { exact zero_lt_succ n },
+      { intro k Hk0 Hkn, apply spectral_sequence.is_contr_E,
+        apply is_contr_ordinary_cohomology,
+        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* }},
+  end
+--   (λm, short_exact_mod_isomorphism
+--     _
+--     isomorphism.rfl
+--     _
+--     (short_exact_mod_of_is_contr_submodules
+--       (convergent_spectral_sequence.HDinf X _)
+--       (zero_lt_succ n)
+--       _))
+
+end cohomology