lean2/hott/homotopy/homotopy_group.hlean

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/-
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 algebra.homotopy_group .sphere
open eq is_trunc trunc_index pointed algebra trunc nat is_conn fiber pointed
namespace is_trunc
-- Lemma 8.3.1
theorem trivial_homotopy_group_of_is_trunc (A : Type*) (n k : ) [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 _,
rewrite [succ_sub_two_succ k],
exact of_nat_le_of_nat H,
end
-- Lemma 8.3.2
theorem trivial_homotopy_group_of_is_conn (A : Type*) {k 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_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
section
open sphere sphere.ops sphere_index
theorem homotopy_group_sphere_le (n k : ) (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
end
theorem is_contr_HG_fiber_of_is_connected {A B : Type*} (k 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)
end is_trunc