Spectral/homotopy/sec83.hlean

-- Section 8.3

import types.trunc types.pointed homotopy.connectedness homotopy.sphere homotopy.circle algebra.group algebra.homotopy_group

open eq is_trunc is_equiv nat equiv trunc function circle algebra pointed trunc_index homotopy


-- 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 (πg[k+1] A) :=
begin
  apply is_trunc_trunc_of_is_trunc,
  apply is_contr_loop_of_is_trunc,
  apply @is_trunc_of_le A n _,
  exact of_nat_le_of_nat H
end

-- 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 H2 : of_nat k ≤ of_nat n, from of_nat_le_of_nat H,
    have H3 : is_contr (ptrunc k A),
    begin
      fapply is_contr_equiv_closed,
      { apply trunc_trunc_equiv_left _ k n H2}
    end,
    have H4 : is_contr (Ω[k](ptrunc k A)),
      from !is_trunc_loop_of_is_trunc,
    apply is_trunc_equiv_closed_rev,
    { apply equiv_of_pequiv (phomotopy_group_pequiv_loop_ptrunc k A)}
end

-- Corollary 8.3.3
open sphere.ops sphere_index
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),
    rewrite [-trunc_index.of_sphere_index_of_nat, -trunc_index.succ_sub_one], apply is_conn_sphere}
end