Created sec83.hlean

homotopy/sec83.hlean

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+-- 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 is_trunc.trunc_index homotopy
+
+notation `Floris` := sorry
+
+-- Lemma 8.3.1
+
+definition homotopy_group_of_is_trunc (A : Type*) (n : ℕ) (p : is_trunc n A) : ∀(k : ℕ), πG[n+k+1] A = G0 :=
+begin
+  intro k,
+  apply @trivial_group_of_is_contr,
+  apply is_trunc_trunc_of_is_trunc,
+  apply is_contr_loop_of_is_trunc,
+  apply @is_trunc_of_leq A n _,
+  induction k with k IHk,
+  {
+    apply is_trunc.trunc_index.le.refl
+  },
+  {
+    induction n with n IHn,
+    {
+      constructor
+    },
+    {
+      exact Floris
+    }
+  }
+end
+
+-- Lemma 8.3.2
+definition trunc_trunc (n k : ℕ₋₂) (p : k ≤ n) (A : Type)
+  : trunc k (trunc n A) ≃ trunc k A :=
+sorry
+
+definition zero_trunc_of_iterated_loop_space (k : ℕ) (A : Type*)
+  : trunc 0 (Ω[k] A) ≃ Ω[k](pointed.MK (trunc k A) (tr pt)) := 
+sorry
+
+
+definition homotopy_group_of_is_conn (A : Type*) (n : ℕ) (p : is_conn n A) : ∀(k : ℕ), (k ≤ n) → is_contr(π[k] A) :=
+begin
+  intros k H,
+  exact Floris
+end
+
+-- Corollary 8.3.3