49 lines
1.2 KiB
Text
49 lines
1.2 KiB
Text
-- Section 8.3
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import types.trunc types.pointed homotopy.connectedness homotopy.sphere homotopy.circle algebra.group algebra.homotopy_group
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open eq is_trunc is_equiv nat equiv trunc function circle algebra pointed is_trunc.trunc_index homotopy
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notation `Floris` := sorry
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-- Lemma 8.3.1
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definition homotopy_group_of_is_trunc (A : Type*) (n : ℕ) (p : is_trunc n A) : ∀(k : ℕ), πG[n+k+1] A = G0 :=
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begin
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intro k,
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apply @trivial_group_of_is_contr,
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apply is_trunc_trunc_of_is_trunc,
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apply is_contr_loop_of_is_trunc,
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apply @is_trunc_of_leq A n _,
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induction k with k IHk,
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{
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apply is_trunc.trunc_index.le.refl
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},
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{
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induction n with n IHn,
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{
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constructor
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},
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{
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exact Floris
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}
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}
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end
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-- Lemma 8.3.2
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definition trunc_trunc (n k : ℕ₋₂) (p : k ≤ n) (A : Type)
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: trunc k (trunc n A) ≃ trunc k A :=
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sorry
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definition zero_trunc_of_iterated_loop_space (k : ℕ) (A : Type*)
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: trunc 0 (Ω[k] A) ≃ Ω[k](pointed.MK (trunc k A) (tr pt)) :=
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sorry
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definition homotopy_group_of_is_conn (A : Type*) (n : ℕ) (p : is_conn n A) : ∀(k : ℕ), (k ≤ n) → is_contr(π[k] A) :=
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begin
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intros k H,
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exact Floris
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end
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-- Corollary 8.3.3
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