ddef24223b
All HITs which automatically have a point are pointed without a 'p' in front. HITs which do not automatically have a point do still have a p (e.g. pushout/ppushout).
There were a lot of annoyances with spheres being indexed by N_{-1} with almost no extra generality. We now index the spheres by nat, making sphere 0 = pbool.
52 lines
1.6 KiB
Text
52 lines
1.6 KiB
Text
/-
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Copyright (c) 2016 Ulrik Buchholtz and Egbert Rijke. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Ulrik Buchholtz, Egbert Rijke, Floris van Doorn
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The H-space structure on S¹ and the complex Hopf fibration
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(the standard one).
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-/
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import .hopf .circle types.fin
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open eq equiv is_equiv circle is_conn trunc is_trunc sphere susp pointed fiber sphere.ops function
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join
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namespace hopf
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definition circle_h_space [instance] : h_space S¹ :=
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⦃ h_space, one := base, mul := circle_mul,
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one_mul := circle_base_mul, mul_one := circle_mul_base ⦄
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definition circle_assoc (x y z : S¹) : (x * y) * z = x * (y * z) :=
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begin
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induction x,
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{ reflexivity },
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{ apply eq_pathover, induction y,
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{ exact natural_square
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(λa : S¹, ap (λb : S¹, b * z) (circle_mul_base a))
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loop },
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{ apply is_prop.elimo, apply is_trunc_square }}
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end
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definition complex_hopf' : S 3 → S 2 :=
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begin
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intro x, apply @sigma.pr1 (susp S¹) (hopf S¹),
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apply inv (hopf.total S¹), exact (join_sphere 1 1)⁻¹ᵉ x
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end
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definition complex_hopf [constructor] : S 3 →* S 2 :=
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proof pmap.mk complex_hopf' idp qed
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definition pfiber_complex_hopf : pfiber complex_hopf ≃* S 1 :=
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begin
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fapply pequiv_of_equiv,
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{ esimp, unfold [complex_hopf'],
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refine fiber.equiv_precompose (sigma.pr1 ∘ (hopf.total S¹)⁻¹ᵉ)
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(join_sphere 1 1)⁻¹ᵉ _ ⬝e _,
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refine fiber.equiv_precompose _ (hopf.total S¹)⁻¹ᵉ _ ⬝e _,
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apply fiber_pr1 },
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{ reflexivity }
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end
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end hopf
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