lean2/hott/homotopy/homotopy_group.hlean

/-
Copyright (c) 2016 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Clive Newstead

-/

import .LES_of_homotopy_groups .sphere .complex_hopf

open eq is_trunc trunc_index pointed algebra trunc nat is_conn fiber pointed unit group

namespace is_trunc

  -- Lemma 8.3.1
  theorem trivial_homotopy_group_of_is_trunc (A : Type*) {n k : ℕ} [is_trunc n A] (H : n < k)
    : is_contr (π[k] A) :=
  begin
    apply is_trunc_trunc_of_is_trunc,
    apply is_contr_loop_of_is_trunc,
    refine @is_trunc_of_le A n _ _ _,
    apply trunc_index.le_of_succ_le_succ,
    rewrite [succ_sub_two_succ k],
    exact of_nat_le_of_nat H,
  end

  theorem trivial_ghomotopy_group_of_is_trunc (A : Type*) (n k : ℕ) [is_trunc n A] (H : n ≤ k)
    : is_contr (πg[k+1] A) :=
  trivial_homotopy_group_of_is_trunc A (lt_succ_of_le H)

  -- Lemma 8.3.2
  theorem trivial_homotopy_group_of_is_conn (A : Type*) {k n : ℕ} (H : k ≤ n) [is_conn n A]
    : is_contr (π[k] A) :=
  begin
    have H3 : is_contr (ptrunc k A), from is_conn_of_le A (of_nat_le_of_nat H),
    have H4 : is_contr (Ω[k](ptrunc k A)), from !is_trunc_loopn_of_is_trunc,
    exact is_trunc_equiv_closed_rev _ (equiv_of_pequiv (homotopy_group_pequiv_loop_ptrunc k A)) _
  end

  -- Corollary 8.3.3
  open sphere sphere.ops
  theorem homotopy_group_sphere_le (n k : ℕ) (H : k < n) : is_contr (π[k] (S n)) :=
  begin
    cases n with n,
    { exfalso, apply not_lt_zero, exact H},
    { have H2 : k ≤ n, from le_of_lt_succ H,
      apply @(trivial_homotopy_group_of_is_conn _ H2) }
  end

  theorem is_contr_HG_fiber_of_is_connected {A B : Type*} (k n : ℕ) (f : A →* B)
    [H : is_conn_fun n f] (H2 : k ≤ n) : is_contr (π[k] (pfiber f)) :=
  @(trivial_homotopy_group_of_is_conn (pfiber f) H2) (H pt)

  /- Corollaries of the LES of homotopy groups -/
  local attribute ab_group.to_group [coercion]
  local attribute is_equiv_tinverse [instance]
  open prod chain_complex group fin equiv function is_equiv lift

  /-
    Because of the construction of the LES this proof only gives us this result when
    A and B live in the same universe (because Lean doesn't have universe cumulativity).
    However, below we also proof that it holds for A and B in arbitrary universes.
  -/
  theorem is_equiv_π_of_is_connected'.{u} {A B : pType.{u}} {n k : ℕ} (f : A →* B)
     (H2 : k ≤ n) [H : is_conn_fun n f] : is_equiv (π→[k] f) :=
  begin
    cases k with k,
    { /- k = 0 -/
      change (is_equiv (trunc_functor 0 f)), apply is_equiv_trunc_functor_of_is_conn_fun,
      refine is_conn_fun_of_le f (zero_le_of_nat n)},
    { /- k > 0 -/
     have H2' : k ≤ n, from le.trans !self_le_succ H2,
     exact LES_is_equiv_of_trivial f (succ k) 0
             (@is_contr_HG_fiber_of_is_connected A B k n f H H2')
             (@is_contr_HG_fiber_of_is_connected A B (succ k) n f H H2) },
  end

  theorem is_equiv_π_of_is_connected.{u v} {A : pType.{u}} {B : pType.{v}} {n k : ℕ} (f : A →* B)
    (H2 : k ≤ n) [H : is_conn_fun n f] : is_equiv (π→[k] f) :=
  begin
    have π→[k] pdown.{v u} ∘* π→[k] (plift_functor f) ∘* π→[k] pup.{u v} ~* π→[k] f,
    begin
      refine pwhisker_left _ !homotopy_group_functor_pcompose⁻¹* ⬝* _,
      refine !homotopy_group_functor_pcompose⁻¹* ⬝* _,
      apply homotopy_group_functor_phomotopy, apply plift_functor_phomotopy
    end,
    have π→[k] pdown.{v u} ∘ π→[k] (plift_functor f) ∘ π→[k] pup.{u v} ~ π→[k] f, from this,
    apply is_equiv.homotopy_closed,
    { exact this},
    { do 2 apply is_equiv_compose,
      { apply is_equiv_homotopy_group_functor, apply to_is_equiv !equiv_lift},
      { refine @(is_equiv_π_of_is_connected' _ H2) _, apply is_conn_fun_lift_functor},
      { apply is_equiv_homotopy_group_functor, apply to_is_equiv !equiv_lift⁻¹ᵉ}}
  end

  definition π_equiv_π_of_is_connected {A B : Type*} {n k : ℕ} (f : A →* B)
     (H2 : k ≤ n) [H : is_conn_fun n f] : π[k] A ≃* π[k] B :=
  pequiv_of_pmap (π→[k] f) (is_equiv_π_of_is_connected f H2)

  -- TODO: prove this for A and B in different universe levels
  theorem is_surjective_π_of_is_connected.{u} {A B : pType.{u}} (n : ℕ) (f : A →* B)
    [H : is_conn_fun n f] : is_surjective (π→[n + 1] f) :=
  @is_surjective_of_trivial _
    (LES_of_homotopy_groups f) _
    (is_exact_LES_of_homotopy_groups f (n, 2))
    (@is_contr_HG_fiber_of_is_connected A B n n f H !le.refl)

  /-
    Theorem 8.8.3: Whitehead's principle and its corollaries
  -/
  definition whitehead_principle (n : ℕ₋₂) {A B : Type}
    [HA : is_trunc n A] [HB : is_trunc n B] (f : A → B) (H' : is_equiv (trunc_functor 0 f))
    (H : Πa k, is_equiv (π→[k + 1] (pmap_of_map f a))) : is_equiv f :=
  begin
    revert A B HA HB f H' H, induction n with n IH: intros,
    { exact is_equiv_of_is_contr _ _ _ },
    have Πa, is_equiv (Ω→ (pmap_of_map f a)),
    begin
      intro a,
      apply IH, do 2 (esimp; exact _),
      { rexact H a 0},
      intro p k,
      have is_equiv (π→[k + 1] (Ω→(pmap_of_map f a))),
        from is_equiv_homotopy_group_functor_ap1 (k+1) (pmap_of_map f a),
      have Π(b : A) (p : a = b),
        is_equiv (pmap.to_fun (π→[k + 1] (pmap_of_map (ap f) p))),
      begin
        intro b p, induction p, refine is_equiv.homotopy_closed _ _ this,
        refine homotopy_group_functor_phomotopy _ _,
        apply ap1_pmap_of_map
      end,
      have is_equiv (homotopy_group_pequiv _
                      (pequiv_of_eq_pt (!idp_con⁻¹ : ap f p = Ω→ (pmap_of_map f a) p)) ∘
           pmap.to_fun (π→[k + 1] (pmap_of_map (ap f) p))),
      begin
        apply is_equiv_compose, exact this a p,
        exact is_equiv_trunc_functor _ _ _
      end,
      refine is_equiv.homotopy_closed _ _ this,
      refine !homotopy_group_functor_pcompose⁻¹* ⬝* _,
      apply homotopy_group_functor_phomotopy,
      fapply phomotopy.mk,
      { esimp, intro q, refine !idp_con⁻¹},
      { esimp, refine !idp_con⁻¹},
    end,
    apply is_equiv_of_is_equiv_ap1_of_is_equiv_trunc
  end

  definition whitehead_principle_pointed (n : ℕ₋₂) {A B : Type*}
    [HA : is_trunc n A] [HB : is_trunc n B] [is_conn 0 A] (f : A →* B)
    (H : Πk, is_equiv (π→[k] f)) : is_equiv f :=
  begin
    apply whitehead_principle n, rexact H 0,
    intro a k, revert a, apply is_conn.elim -1,
    have is_equiv (π→[k + 1] (pointed_eta_pequiv B ⬝e* (pequiv_of_eq_pt (respect_pt f))⁻¹ᵉ*)
           ∘* π→[k + 1] f ∘* π→[k + 1] (pointed_eta_pequiv A)⁻¹ᵉ*),
    begin
      apply is_equiv_compose
              (π→[k + 1] (pointed_eta_pequiv B ⬝e* (pequiv_of_eq_pt (respect_pt f))⁻¹ᵉ*)),
      refine is_equiv_compose (π→[k + 1] f) _ _ _,
      all_goals exact is_equiv_homotopy_group_functor _ _ _,
    end,
    refine is_equiv.homotopy_closed _ _ this,
    apply to_homotopy,
    refine pwhisker_left _ !homotopy_group_functor_pcompose⁻¹* ⬝* _,
    refine !homotopy_group_functor_pcompose⁻¹* ⬝* _,
    apply homotopy_group_functor_phomotopy, apply phomotopy_pmap_of_map
  end

  open pointed.ops
  definition is_contr_of_trivial_homotopy (n : ℕ₋₂) (A : Type) [is_trunc n A] [is_conn 0 A]
    (H : Πk a, is_contr (π[k] (pointed.MK A a))) : is_contr A :=
  begin
    refine is_trunc_is_equiv_closed_rev _ (λa, ⋆) _ _,
    apply whitehead_principle n,
    { apply is_equiv_trunc_functor_of_is_conn_fun, apply is_conn_fun_to_unit_of_is_conn },
    intro a k,
    refine is_equiv_of_is_contr _ _ _,
    exact trivial_homotopy_group_of_is_trunc _ !zero_lt_succ,
  end

  definition is_contr_of_trivial_homotopy_nat (n : ℕ) (A : Type) [is_trunc n A] [is_conn 0 A]
    (H : Πk a, k ≤ n → is_contr (π[k] (pointed.MK A a))) : is_contr A :=
  begin
    apply is_contr_of_trivial_homotopy n,
    intro k a, apply @lt_ge_by_cases _ _ n k,
    { intro H', exact trivial_homotopy_group_of_is_trunc _ H'},
    { intro H', exact H k a H'}
  end

  definition is_contr_of_trivial_homotopy_pointed (n : ℕ₋₂) (A : Type*) [is_trunc n A]
    (H : Πk, is_contr (π[k] A)) : is_contr A :=
  begin
    have is_conn 0 A, proof H 0 qed,
    fapply is_contr_of_trivial_homotopy n A,
    intro k, apply is_conn.elim -1,
    cases A with A a, exact H k
  end

  definition is_contr_of_trivial_homotopy_nat_pointed (n : ℕ) (A : Type*) [is_trunc n A]
    (H : Πk, k ≤ n → is_contr (π[k] A)) : is_contr A :=
  begin
    have is_conn 0 A, proof H 0 !zero_le qed,
    fapply is_contr_of_trivial_homotopy_nat n A,
    intro k a H', revert a, apply is_conn.elim -1,
    cases A with A a, exact H k H'
  end

  definition is_trunc_of_trivial_homotopy {n : ℕ} {m : ℕ₋₂} {A : Type} (H : is_trunc m A)
    (H2 : Πk a, k > n → is_contr (π[k] (pointed.MK A a))) : is_trunc n A :=
  begin
    refine is_trunc_is_equiv_closed_rev _ (@tr n A) _ _,
    apply whitehead_principle m,
    { apply is_equiv_trunc_functor_of_is_conn_fun,
      note z := is_conn_fun_tr n A,
      apply is_conn_fun_of_le _ (of_nat_le_of_nat (zero_le n)), },
    intro a k,
    apply @nat.lt_ge_by_cases n (k+1),
    { intro H3, apply @is_equiv_of_is_contr, exact H2 _ _ H3,
      refine @trivial_homotopy_group_of_is_trunc _ _ _ _ H3 },
    { intro H3, apply @(is_equiv_π_of_is_connected _ H3), apply is_conn_fun_tr }
  end

  definition is_trunc_of_trivial_homotopy_pointed {n : ℕ} {m : ℕ₋₂} {A : Type*} (H : is_trunc m A)
    (Hconn : is_conn 0 A) (H2 : Πk, k > n → is_contr (π[k] A)) : is_trunc n A :=
  begin
    apply is_trunc_of_trivial_homotopy H,
    intro k a H3, revert a, apply is_conn.elim -1,
    cases A with A a₀, exact H2 k H3
  end

  definition is_trunc_of_is_trunc_succ {n : ℕ} {A : Type} (H : is_trunc (n.+1) A)
    (H2 : Πa, is_contr (π[n+1] (pointed.MK A a))) : is_trunc n A :=
  begin
    apply is_trunc_of_trivial_homotopy H,
    intro k a H3, induction H3 with k H3 IH, exact H2 a,
    apply @trivial_homotopy_group_of_is_trunc _ (n+1) _ H, exact succ_le_succ H3
  end

  definition is_trunc_of_is_trunc_succ_pointed {n : ℕ} {A : Type*} (H : is_trunc (n.+1) A)
    (Hconn : is_conn 0 A) (H2 : is_contr (π[n+1] A)) : is_trunc n A :=
  begin
    apply is_trunc_of_trivial_homotopy_pointed H Hconn,
    intro k H3, induction H3 with k H3 IH, exact H2,
    apply @trivial_homotopy_group_of_is_trunc _ (n+1) _ H, exact succ_le_succ H3
  end

  definition ab_group_homotopy_group_of_is_conn (n : ℕ) (A : Type*) [H : is_conn 1 A] :
    ab_group (π[n] A) :=
  begin
    have is_conn 0 A, from !is_conn_of_is_conn_succ,
    cases n with n,
    { unfold [homotopy_group, ptrunc], exact ab_group_of_is_contr _ _ },
    cases n with n,
    { unfold [homotopy_group, ptrunc], exact ab_group_of_is_contr _ _ },
    exact ab_group_homotopy_group (n+2) A
  end

  definition is_contr_of_trivial_homotopy' (n : ℕ₋₂) (A : Type) [is_trunc n A] [is_conn -1 A]
    (H : Πk a, is_contr (π[k] (pointed.MK A a))) : is_contr A :=
  begin
    assert aa : trunc -1 A,
    { apply center },
    assert H3 : is_conn 0 A,
    { induction aa with a, exact H 0 a },
    exact is_contr_of_trivial_homotopy n A H
  end

  definition is_conn_of_trivial_homotopy (n : ℕ₋₂) (m : ℕ) (A : Type) [is_trunc n A] [is_conn 0 A]
    (H : Π(k : ℕ) a, k ≤ m → is_contr (π[k] (pointed.MK A a))) : is_conn m A :=
  begin
    apply is_contr_of_trivial_homotopy_nat m (trunc m A),
    intro k a H2,
    induction a with a,
    apply is_trunc_equiv_closed_rev,
      exact equiv_of_pequiv (homotopy_group_ptrunc_of_le H2 (pointed.MK A a)),
    exact H k a H2
  end

  definition is_conn_of_trivial_homotopy_pointed (n : ℕ₋₂) (m : ℕ) (A : Type*) [is_trunc n A]
    (H : Π(k : ℕ), k ≤ m → is_contr (π[k] A)) : is_conn m A :=
  begin
    have is_conn 0 A, proof H 0 !zero_le qed,
    apply is_conn_of_trivial_homotopy n m A,
    intro k a H2, revert a, apply is_conn.elim -1,
    cases A with A a, exact H k H2
  end

  definition is_conn_fun_of_equiv_on_homotopy_groups.{u} (n : ℕ) {A B : Type.{u}} (f : A → B)
    [is_equiv (trunc_functor 0 f)]
    (H1 : Πa k, k ≤ n → is_equiv (homotopy_group_functor k (pmap_of_map f a)))
    (H2 : Πa, is_surjective (homotopy_group_functor (succ n) (pmap_of_map f a))) : is_conn_fun n f :=
  have H2' : Πa k, k ≤ n → is_surjective (homotopy_group_functor (succ k) (pmap_of_map f a)),
  begin
    intro a k H, cases H with n' H',
    { apply H2},
    { apply is_surjective_of_is_equiv, apply H1, exact succ_le_succ H'}
  end,
  have H3 : Πa, is_contr (ptrunc n (pfiber (pmap_of_map f a))),
  begin
    intro a, apply is_contr_of_trivial_homotopy_nat_pointed n,
    { intro k H, apply is_trunc_equiv_closed_rev, exact homotopy_group_ptrunc_of_le H _,
      rexact @is_contr_of_is_embedding_of_is_surjective +3ℕ
               (LES_of_homotopy_groups (pmap_of_map f a)) (k, 0)
               (is_exact_LES_of_homotopy_groups _ _)
               proof @(is_embedding_of_is_equiv _)  (H1 a k H) qed
               proof (H2' a k H) qed}
  end,
  show Πb, is_contr (trunc n (fiber f b)),
  begin
    intro b,
    note p := right_inv (trunc_functor 0 f) (tr b), revert p,
    induction (trunc_functor 0 f)⁻¹ (tr b), esimp, intro p,
    induction !tr_eq_tr_equiv p with q,
    rewrite -q, exact H3 a
  end

end is_trunc
open is_trunc function
/- applications to infty-connected types and maps -/
namespace is_conn

  definition is_conn_fun_inf_of_equiv_on_homotopy_groups.{u} {A B : Type.{u}} (f : A → B)
    [is_equiv (trunc_functor 0 f)]
    (H1 : Πa k, is_equiv (homotopy_group_functor k (pmap_of_map f a))) : is_conn_fun_inf f :=
  begin
    apply is_conn_fun_inf.mk_nat, intro n, apply is_conn_fun_of_equiv_on_homotopy_groups,
    { intro a k H, exact H1 a k},
    { intro a, apply is_surjective_of_is_equiv}
  end

  definition is_equiv_trunc_functor_of_is_conn_fun_inf.{u} (n : ℕ₋₂) {A B : Type.{u}} (f : A → B)
    [is_conn_fun_inf f] : is_equiv (trunc_functor n f) :=
  _

  definition is_equiv_homotopy_group_functor_of_is_conn_fun_inf.{u} {A B : pType.{u}} (f : A →* B)
    [is_conn_fun_inf f] (a : A) (k : ℕ) : is_equiv (homotopy_group_functor k f) :=
  is_equiv_π_of_is_connected f (le.refl k)

end is_conn