37fbe56b8c
The maps on every level are just the functorial action of the homotopy groups (possibly composed by a cast), but there are no compositions with path inversion. There are also some updates in various files after changes in the HoTT library.
125 lines
4.9 KiB
Text
125 lines
4.9 KiB
Text
/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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-/
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import .LES_of_homotopy_groups homotopy.connectedness homotopy.homotopy_group homotopy.join
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open eq is_trunc pointed is_conn is_equiv fiber equiv trunc nat chain_complex prod fin algebra
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group trunc_index function join pushout
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namespace nat
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open sigma sum
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definition eq_even_or_eq_odd (n : ℕ) : (Σk, 2 * k = n) ⊎ (Σk, 2 * k + 1 = n) :=
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begin
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induction n with n IH,
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{ exact inl ⟨0, idp⟩},
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{ induction IH with H H: induction H with k p: induction p,
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{ exact inr ⟨k, idp⟩},
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{ refine inl ⟨k+1, idp⟩}}
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end
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definition rec_on_even_odd {P : ℕ → Type} (n : ℕ) (H : Πk, P (2 * k)) (H2 : Πk, P (2 * k + 1))
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: P n :=
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begin
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cases eq_even_or_eq_odd n with v v: induction v with k p: induction p,
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{ exact H k},
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{ exact H2 k}
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end
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end nat
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open nat
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namespace is_conn
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local attribute comm_group.to_group [coercion]
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local attribute is_equiv_tinverse [instance]
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theorem is_equiv_π_of_is_connected.{u} {A B : pType.{u}} (n k : ℕ) (f : A →* B)
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[H : is_conn_fun n f] (H2 : k ≤ n) : is_equiv (π→[k] f) :=
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begin
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induction k using rec_on_even_odd with k: cases k with k,
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{ /- k = 0 -/
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change (is_equiv (trunc_functor 0 f)), apply is_equiv_trunc_functor_of_is_conn_fun,
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refine is_conn_fun_of_le f (zero_le_of_nat n)},
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{ /- k > 0 even -/
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have H2' : 2 * k + 1 ≤ n, from le.trans !self_le_succ H2,
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exact
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@is_equiv_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (k, 5))
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(is_exact_LES_of_homotopy_groups3 f (succ k, 0))
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(@is_contr_HG_fiber_of_is_connected A B (2 * k + 1) n f H H2')
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(@is_contr_HG_fiber_of_is_connected A B (2 * succ k) n f H H2)
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(@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 0) idp)
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(@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 1) idp)
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(homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun3 f (k, 0)))},
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{ /- k = 1 -/
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exact sorry}, -- need some more facts about anti-homomorphisms
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{ /- k > 1 odd -/
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have H2' : 2 * succ k ≤ n, from le.trans !self_le_succ H2,
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have H3 : is_equiv (π→*[2*(succ k) + 1] f ∘* tinverse), from
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@is_equiv_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (succ k, 2))
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(is_exact_LES_of_homotopy_groups3 f (succ k, 3))
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(@is_contr_HG_fiber_of_is_connected A B (2 * succ k) n f H H2')
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(@is_contr_HG_fiber_of_is_connected A B (2 * succ k + 1) n f H H2)
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(@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 3) idp)
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(@pgroup_of_group _ (comm_group_LES_of_homotopy_groups3 f k 4) idp)
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(homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun3 f (k, 3))),
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exact @(is_equiv.cancel_right tinverse) !is_equiv_tinverse
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(pmap.to_fun (π→*[2*(succ k) + 1] f)) H3}
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end
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theorem is_surjective_π_of_is_connected.{u} {A B : pType.{u}} (n : ℕ) (f : A →* B)
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[H : is_conn_fun n f] : is_surjective (π→[n + 1] f) :=
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begin
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induction n using rec_on_even_odd with n,
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{ have H3 : is_surjective (π→*[2*n + 1] f ∘* tinverse), from
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@is_surjective_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (n, 2))
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(@is_contr_HG_fiber_of_is_connected A B (2 * n) (2 * n) f H !le.refl),
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exact @(is_surjective_cancel_right (pmap.to_fun (π→*[2*n + 1] f)) tinverse) H3},
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{ exact @is_surjective_of_trivial _
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(LES_of_homotopy_groups3 f) _
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(is_exact_LES_of_homotopy_groups3 f (k, 5))
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(@is_contr_HG_fiber_of_is_connected A B (2 * k + 1) (2 * k + 1) f H !le.refl)}
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end
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/- joins -/
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definition join_empty_right [constructor] (A : Type) : join A empty ≃ A :=
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begin
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fapply equiv.MK,
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{ intro x, induction x with a o a o,
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{ exact a },
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{ exact empty.elim o },
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{ exact empty.elim o } },
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{ exact pushout.inl },
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{ intro a, reflexivity},
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{ intro x, induction x with a o a o,
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{ reflexivity },
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{ exact empty.elim o },
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{ exact empty.elim o } }
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end
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definition natural_square2 {A B X : Type} {f : A → X} {g : B → X} (h : Πa b, f a = g b)
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{a a' : A} {b b' : B} (p : a = a') (q : b = b')
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: square (ap f p) (ap g q) (h a b) (h a' b') :=
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by induction p; induction q; exact hrfl
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section
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open sphere sphere_index
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definition add_plus_one_minus_one (n : ℕ₋₁) : n +1+ -1 = n := idp
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definition add_plus_one_succ (n m : ℕ₋₁) : n +1+ (m.+1) = (n +1+ m).+1 := idp
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definition minus_one_add_plus_one (n : ℕ₋₁) : -1 +1+ n = n :=
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begin induction n with n IH, reflexivity, exact ap succ IH end
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definition succ_add_plus_one (n m : ℕ₋₁) : (n.+1) +1+ m = (n +1+ m).+1 :=
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begin induction m with m IH, reflexivity, exact ap succ IH end
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end
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end is_conn
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