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π₄S³≡ℤ/2 #715

Merged
merged 8 commits into from
Feb 8, 2022
Merged

π₄S³≡ℤ/2 #715

merged 8 commits into from
Feb 8, 2022

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aljungstrom
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@aljungstrom aljungstrom commented Feb 8, 2022

Finally managed to clean this mess up a bit... This PR contains the final steps of the proof of π₄S³≡ℤ/2. More specifically:

  1. A rather silly construction of exact sequences of 4 groups (Exact4). This construction was pretty useful for transporting between various exact sequences, so I think we should keep it. It captures a pretty common situation anyway.
  2. A proof that given such a sequence ℤ -ᶠ→ ℤ → A → 1, we get an iso A ≅ ℤ/abs(f(1)).
  3. The construction of the iso π₄S³≅ℤ/n, where n is some complicated number defined as the Hopf invariant of the Whitehead product of etc. etc. (Cubical.Homotopy.Group.Pi4S3.BrunerieIso)
  4. A proof that this number is 2 Cubical.Homotopy.HopfInvariant.Brunerie.
  5. The final result (Cubical.Homotopy.Group.Pi4S3.Summary)

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@mortberg mortberg left a comment

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This is amazing! I had a lot of small nitpicking comments and the only major change I would like to do is to rewrite the Summary a bit

Cubical/Algebra/Group/Instances/IntMod.agda Outdated Show resolved Hide resolved
Cubical/Algebra/Group/IsomorphismTheorems.agda Outdated Show resolved Hide resolved
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Cubical/Homotopy/Group/Pi4S3/BrunerieIso.agda Outdated Show resolved Hide resolved
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@@ -4,13 +4,16 @@ This file contains a summary of what remains for π₄(S³) ≡ ℤ/2ℤ to be p

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The comment above here needs to be updated

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Actually I think the file should be reorganized a bit, maybe dropping the π₄S³ module and just stating the key results in sequence

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@aljungstrom
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@mortberg So, fixed the stuff now. As discussed I left the summary file for you:-)

@mortberg mortberg merged commit f39fb72 into agda:master Feb 8, 2022
@kangxyz
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kangxyz commented Feb 8, 2022

Congratulations! It's great to see a complete formalization come into reality!

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3 participants