92 lines
3.5 KiB
Text
92 lines
3.5 KiB
Text
/-
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Copyright (c) 2016 Floris van Doorn. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Floris van Doorn
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Calculating homotopy groups of spheres.
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In this file we calculate
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π₂(S²) = Z
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πₙ(S²) = πₙ(S³) for n > 2
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πₙ(Sⁿ) = Z for n > 0
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π₂(S³) = Z
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-/
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import .homotopy_group .freudenthal
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open eq group algebra is_equiv equiv fin prod chain_complex pointed fiber nat is_trunc trunc_index
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sphere.ops trunc is_conn susp bool
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namespace sphere
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/- Corollaries of the complex hopf fibration combined with the LES of homotopy groups -/
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open sphere sphere.ops int circle hopf
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definition π2S2 : πg[2] (S 2) ≃g gℤ :=
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begin
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refine _ ⬝g fundamental_group_of_circle,
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refine _ ⬝g homotopy_group_isomorphism_of_pequiv _ pfiber_complex_hopf,
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fapply isomorphism_of_equiv,
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{ fapply equiv.mk,
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{ exact cc_to_fn (LES_of_homotopy_groups complex_hopf) (1, 2)},
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{ refine LES_is_equiv_of_trivial complex_hopf 1 2 _ _,
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{ have H : 1 ≤[ℕ] 2, from !one_le_succ,
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apply trivial_homotopy_group_of_is_conn, exact H, rexact is_conn_sphere 3 },
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{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
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(LES_of_homotopy_groups_1 complex_hopf 2) _,
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apply trivial_homotopy_group_of_is_conn, apply le.refl, rexact is_conn_sphere 3 }}},
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{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (0, 2))}
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end
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open circle
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definition πnS3_eq_πnS2 (n : ℕ) : πg[n+3] (S 3) ≃g πg[n+3] (S 2) :=
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begin
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fapply isomorphism_of_equiv,
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{ fapply equiv.mk,
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{ exact cc_to_fn (LES_of_homotopy_groups complex_hopf) (n+3, 0)},
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{ have H : is_trunc 1 (pfiber complex_hopf),
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from @(is_trunc_equiv_closed_rev _ pfiber_complex_hopf) is_trunc_circle,
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refine LES_is_equiv_of_trivial complex_hopf (n+3) 0 _ _,
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{ have H2 : 1 ≤[ℕ] n + 1, from !one_le_succ,
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exact @trivial_ghomotopy_group_of_is_trunc _ _ _ H H2 },
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{ refine tr_rev (λx, is_contr (ptrunctype._trans_of_to_pType x))
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(LES_of_homotopy_groups_2 complex_hopf _) _,
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have H2 : 1 ≤[ℕ] n + 2, from !one_le_succ,
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apply trivial_ghomotopy_group_of_is_trunc _ _ _ H2 }}},
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{ exact homomorphism.struct (homomorphism_LES_of_homotopy_groups_fun _ (n+2, 0))}
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end
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definition sphere_stability_pequiv (k n : ℕ) (H : k + 2 ≤ 2 * n) :
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π[k + 1] (S (n+1)) ≃* π[k] (S n) :=
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iterate_susp_stability_pequiv pbool H
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definition stability_isomorphism (k n : ℕ) (H : k + 3 ≤ 2 * n)
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: πg[k+1 +1] (S (n+1)) ≃g πg[k+1] (S n) :=
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iterate_susp_stability_isomorphism pbool H
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open int circle hopf
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definition πnSn (n : ℕ) [H : is_succ n] : πg[n] (S (n)) ≃g gℤ :=
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begin
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induction H with n,
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cases n with n IH,
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{ exact fundamental_group_of_circle },
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{ induction n with n IH,
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{ exact π2S2 },
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{ refine _ ⬝g IH, apply stability_isomorphism,
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rexact add_mul_le_mul_add n 1 2 }}
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end
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theorem not_is_trunc_sphere (n : ℕ) : ¬is_trunc n (S (n+1)) :=
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begin
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intro H,
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note H2 := trivial_ghomotopy_group_of_is_trunc (S (n+1)) n n !le.refl,
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have H3 : is_contr ℤ, from is_trunc_equiv_closed _ (equiv_of_isomorphism (πnSn (n+1))),
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have H4 : (0 : ℤ) ≠ (1 : ℤ), from dec_star,
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apply H4,
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apply is_prop.elim,
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end
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definition π3S2 : πg[3] (S 2) ≃g gℤ :=
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begin
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refine _ ⬝g πnSn 3, symmetry, rexact πnS3_eq_πnS2 0
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end
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end sphere
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