12a9345df1
This is still work in progress. Spectral sequences should be more usable, and probably the degrees of graded maps should be group homomorphisms so that we can reindex spectral sequences.
137 lines
5.4 KiB
Text
137 lines
5.4 KiB
Text
/- A computation of the cohomology groups of K(ℤ,2) using the Serre spectral sequence
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Author: Floris van Doorn-/
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import .serre
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open eq spectrum EM EM.ops int pointed cohomology left_module algebra group fiber is_equiv equiv
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prod is_trunc function
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namespace temp
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definition uH0_circle : uH^0[circle] ≃g gℤ :=
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sorry
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definition uH1_circle : uH^1[circle] ≃g gℤ :=
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sorry
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definition uH_circle_of_ge (n : ℤ) (h : n ≥ 2) : uH^n[circle] ≃g trivial_ab_group :=
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sorry
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definition f : unit → K agℤ 2 :=
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λx, pt
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definition fserre :
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(λn s, uoH^-(n-s)[K agℤ 2, H^-s[circle₊]]) ⟹ᵍ (λn, H^-n[unit₊]) :=
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proof
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converges_to_g_isomorphism
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(serre_convergence_map_of_is_conn pt f (EM_spectrum agℤ) 0
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(is_strunc_EM_spectrum agℤ) (is_conn_EM agℤ 2))
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begin
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intro n s, apply unreduced_ordinary_cohomology_isomorphism_right,
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apply unreduced_cohomology_isomorphism, symmetry,
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refine !fiber_const_equiv ⬝e _,
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refine loop_EM _ 1 ⬝e _,
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exact EM_pequiv_circle
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end
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begin intro n, reflexivity end
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qed
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section
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local notation `X` := converges_to.X fserre
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local notation `E∞` := convergence_theorem.Einf (converges_to.HH fserre)
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local notation `E∞d` := convergence_theorem.Einfdiag (converges_to.HH fserre)
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local notation `E` := exact_couple.E X
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definition fbuilt (n : ℤ) :
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is_built_from (LeftModule_int_of_AbGroup (H^-n[unit₊])) (E∞d (n, 0)) :=
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is_built_from_of_converges_to fserre n
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definition fEinf0 : E∞ (0, 0) ≃lm LeftModule_int_of_AbGroup agℤ :=
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isomorphism_zero_of_is_built_from (fbuilt 0) (by reflexivity) ⬝lm
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lm_iso_int.mk (cohomology_change_int _ _ neg_zero ⬝g
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cohomology_isomorphism pbool_pequiv_add_point_unit _ _ ⬝g ordinary_cohomology_pbool _)
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definition fEinfd (n : ℤ) (m : ℕ) (p : n ≠ 0) : is_contr (E∞d (n, 0) m) :=
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have p' : -n ≠ 0, from λH, p (eq_zero_of_neg_eq_zero H),
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is_contr_quotients (fbuilt n) (@(is_trunc_equiv_closed_rev -2
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(group.equiv_of_isomorphism (cohomology_isomorphism pbool_pequiv_add_point_unit _ _)))
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(EM_dimension' _ _ p')) _
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definition fEinf (n : ℤ) (m : ℕ) (p : n ≠ 0) : is_contr (E∞ (n, -m)) :=
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transport (is_contr ∘ E∞)
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begin
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induction m with m q, reflexivity, refine ap (deg (exact_couple.i X)) q ⬝ _,
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exact prod_eq idp (neg_add m 1)⁻¹
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end
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(fEinfd n m p)
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definition is_contr_fD (n s : ℤ) (p : s > 0) : is_contr (E (n, s)) :=
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have is_contr H^-s[circle₊], from
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is_contr_ordinary_cohomology_of_neg _ _ (neg_neg_of_pos p),
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have is_contr (uoH^-(n-s)[K agℤ 2, H^-s[circle₊]]), from
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is_contr_unreduced_ordinary_cohomology _ _ _ _,
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@(is_contr_equiv_closed (left_module.equiv_of_isomorphism (converges_to.e fserre (n, s))⁻¹ˡᵐ))
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this
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definition is_contr_fD2 (n s : ℤ) (p : n > s) : is_contr (E (n, s)) :=
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have -(n-s) < 0, from neg_neg_of_pos (sub_pos_of_lt p),
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@(is_contr_equiv_closed (left_module.equiv_of_isomorphism (converges_to.e fserre (n, s))⁻¹ˡᵐ))
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(is_contr_ordinary_cohomology_of_neg _ _ this)
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definition is_contr_fD3 (n s : ℤ) (p : s ≤ - 2) : is_contr (E (n, s)) :=
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have -s ≥ 2, from sorry, --from neg_neg_of_pos (sub_pos_of_lt p),
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@(is_contr_equiv_closed (group.equiv_of_isomorphism (unreduced_ordinary_cohomology_isomorphism_right _ (uH_circle_of_ge _ this)⁻¹ᵍ _) ⬝e
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left_module.equiv_of_isomorphism (converges_to.e fserre (n, s))⁻¹ˡᵐ))
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(is_contr_ordinary_cohomology _ _ _ !is_contr_unit)
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--(unreduced_ordinary_cohomology_isomorphism_right _ _ _)
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--(is_contr_ordinary_cohomology_of_neg _ _ this)
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--(is_contr_ordinary_cohomology_of_neg _ _ this)
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definition fE00 : E (0,0) ≃lm LeftModule_int_of_AbGroup agℤ :=
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begin
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refine (Einf_isomorphism fserre 0 _ _)⁻¹ˡᵐ ⬝lm fEinf0,
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intro r H, apply is_contr_fD2, exact sub_nat_lt 0 (r+1),
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intro r H, apply is_contr_fD, change 0 + (r + 1) >[ℤ] 0,
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apply of_nat_lt_of_nat_of_lt,
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apply nat.zero_lt_succ,
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end
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definition Ex0 (n : ℕ) : AddGroup_of_AddAbGroup (E (-n,0)) ≃g uH^n[K agℤ 2] :=
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begin
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refine group_isomorphism_of_lm_isomorphism_int (converges_to.e fserre (-n,0)) ⬝g _,
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refine cohomology_change_int _ _ (ap neg !sub_zero ⬝ !neg_neg) ⬝g
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unreduced_ordinary_cohomology_isomorphism_right _ uH0_circle _,
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end
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definition Ex1 (n : ℕ) : AddGroup_of_AddAbGroup (E (-(n+1),- 1)) ≃g uH^n[K agℤ 2] :=
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begin
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refine group_isomorphism_of_lm_isomorphism_int (converges_to.e fserre (-(n+1),- 1)) ⬝g _,
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refine cohomology_change_int _ _ (ap neg _ ⬝ !neg_neg) ⬝g
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unreduced_ordinary_cohomology_isomorphism_right _ !uH1_circle _,
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exact ap (λx, x - - 1) !neg_add ⬝ !add_sub_cancel
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end
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definition uH0 : uH^0[K agℤ 2] ≃g gℤ :=
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(Ex0 0)⁻¹ᵍ ⬝g group_isomorphism_of_lm_isomorphism_int fE00
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definition fE10 : is_contr (E (- 1,0)) :=
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begin
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refine @(is_trunc_equiv_closed _ _) (fEinf (- 1) 0 dec_star),
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apply equiv_of_isomorphism,
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refine Einf_isomorphism fserre 0 _ _,
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intro r H, --apply is_contr_fD2, change (- 1) - (- 1) >[ℤ] (- 0) - (r + 1),
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-- apply is_contr_fD, change (-0) - (r + 1) >[ℤ] 0,
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--exact sub_nat_lt 0 r,
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-- intro r H, apply is_contr_fD, change 0 + (r + 1) >[ℤ] 0,
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-- apply of_nat_lt_of_nat_of_lt,
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-- apply nat.zero_lt_succ,
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end
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definition uH1 : is_contr (uH^1[K agℤ 2]) :=
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begin
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refine @(is_trunc_equiv_closed -2 (group.equiv_of_isomorphism !Ex0)) fE10,
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end
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end
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end temp
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