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feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
/-
Copyright (c) 2016 Ulrik Buchholtz and Egbert Rijke. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
Authors: Ulrik Buchholtz, Egbert Rijke, Floris van Doorn
feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
The H-space structure on S¹ and the complex Hopf fibration
(the standard one).
-/
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
import .hopf .circle types.fin
feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
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open eq equiv is_equiv circle is_conn trunc is_trunc sphere susp pointed fiber sphere.ops function
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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join
feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
namespace hopf
definition circle_h_space [instance] : h_space S¹ :=
⦃ h_space, one := base, mul := circle_mul,
one_mul := circle_base_mul, mul_one := circle_mul_base ⦄
definition circle_assoc (x y z : S¹) : (x * y) * z = x * (y * z) :=
begin
induction x,
{ reflexivity },
{ apply eq_pathover, induction y,
feat(hott) move many lemmas to library, and cleanup various parts
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{ exact natural_square
feat(hott): the complex hopf fibration S3 to S2
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(λa : S¹, ap (λb : S¹, b * z) (circle_mul_base a))
loop },
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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{ apply is_prop.elimo, apply is_trunc_square }}
feat(hott): the complex hopf fibration S3 to S2
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end
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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definition complex_hopf' : S 3 → S 2 :=
feat(hott): the complex hopf fibration S3 to S2
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begin
intro x, apply @sigma.pr1 (susp S¹) (hopf S¹),
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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apply inv (hopf.total S¹), exact (join_sphere 1 1)⁻¹ᵉ x
feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
end
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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definition complex_hopf [constructor] : S 3 →* S 2 :=
proof pmap.mk complex_hopf' idp qed
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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definition pfiber_complex_hopf : pfiber complex_hopf ≃* S 1 :=
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
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begin
fapply pequiv_of_equiv,
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
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{ esimp, unfold [complex_hopf'],
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
refine fiber.equiv_precompose (sigma.pr1 ∘ (hopf.total S¹)⁻¹ᵉ)
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
(join_sphere 1 1)⁻¹ᵉ _ ⬝e _,
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
refine fiber.equiv_precompose _ (hopf.total S¹)⁻¹ᵉ _ ⬝e _,
make pointed suspensions, wedges and spheres the default (in contrast to the unpointed ones), remove sphere_index 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.
2017-07-20 14:01:40 +00:00
apply fiber_pr1 },
{ reflexivity }
feat(hott): Port files from other repositories to the HoTT library. This commit adds truncated 2-quotients, groupoid quotients, Eilenberg MacLane spaces, chain complexes, the long exact sequence of homotopy groups, the Freudenthal Suspension Theorem, Whitehead's principle, and the computation of homotopy groups of almost all spheres which are known in HoTT.
2016-04-22 19:12:25 +00:00
end
feat(hott): the complex hopf fibration S3 to S2
2016-03-06 23:09:32 +00:00
end hopf
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