Discussion of Homotopy Type Theory and Univalent Foundations
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From: "'Urs Schreiber' via Homotopy Type Theory" <HomotopyTypeTheory@googlegroups.com>
To: Homotopy Type Theory <homotopytypetheory@googlegroups.com>
Cc: "Steve Awodey" <awodey@cmu.edu>,
	"Joyal, André" <joyal.andre@uqam.ca>,
	"Anders Mortberg" <andersmortberg@gmail.com>
Subject: Re: [HoTT] Formalization of π₄(S³)≅ℤ/2ℤ in Cubical Agda completed
Date: Wed, 9 Feb 2022 08:44:23 +0400	[thread overview]
Message-ID: <CA+Kbugc-fn9wupcbxTekp4z8aJn3B-OOWc1-1Gpp=GYk9LrmbQ@mail.gmail.com> (raw)
In-Reply-To: <CAO0tDg5V2dMm71XbqR4dt+K2jCZpE-myseR7385H=ZGbpoWytQ@mail.gmail.com>

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Just to add that Pontrjagin had announced the result
(on the second stem) already at the ICM in 1936:

  ncatlab.org/nlab/files/PontrjaginSurLesTransformationDesSpheres.pdf
  ncatlab.org/nlab/show/second+stable+homotopy+group+of+spheres#references

and that the method he used was, of course, his theorem
on identifying Cohomotopy with Cobordism:

 ncatlab.org/nlab/show/Pontryagin+theorem

Two decades later Thom would make a splash by re-inventing (and generalizing)
this method, and it was only then that Pontryagin bothered to write more of
an exposition of his method (references behind the above link).

Best wishes,
urs

On Wed, Feb 9, 2022 at 8:16 AM Noah Snyder <nsnyder@gmail.com> wrote:
>
> I think it goes back to Pontryagin from the late 1930s, though with a rather different argument. The history is a bit confusing because Pontryagin famously made a mistake in computing \pi_4(S^2). But his calculation of \pi_4(S^3) was fine.  See Theorem 2' and 2'' of https://people.math.rochester.edu/faculty/doug/otherpapers/pont2.pdf which I found at this webpage that has more history: https://people.math.rochester.edu/faculty/doug/AKpapers.html#2-stem
>
> Best,
>
> Noah
>
> On Tue, Feb 8, 2022 at 10:09 PM Steve Awodey <awodey@cmu.edu> wrote:
>>
>> Someone please correct me if I am wrong,
>> but I believe that this was originally proved by J-P Serre in his 1951 dissertation (using spectral sequences).
>> So we are about 70 years behind - but catching up fast!
>>
>> Congratulations to all who contributed to this milestone!
>>
>> Univalent regards,
>>
>> Steve
>>
>>
>> On Feb 8, 2022, at 6:24 PM, Joyal, André <joyal.andre@uqam.ca> wrote:
>>
>> Dear Anders,
>>
>> Congratulation to you and Axel for doing this!
>> It was a problem for many years!
>> We can now honestly say that cubical Agda is a serious tool in homotopy theory!
>> It is an amazing piece of work.
>> Thanks to all, starting with Guillaume.
>>
>> Best wishes,
>> Andre
>> ________________________________
>> De : homotopytypetheory@googlegroups.com <homotopytypetheory@googlegroups.com> de la part de Anders Mortberg <andersmortberg@gmail.com>
>> Envoyé : 8 février 2022 15:19
>> À : Homotopy Type Theory <homotopytypetheory@googlegroups.com>
>> Objet : [HoTT] Formalization of π₄(S³)≅ℤ/2ℤ in Cubical Agda completed
>>
>> We are happy to announce that we have finished a formalization of  π₄(S³)≅ℤ/2ℤ  in Cubical Agda. Most of the code has been written by my PhD student Axel Ljungström and the proof largely follows Guillaume Brunerie's PhD thesis. For details and a summary see:
>>
>> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda
>>
>> The proof involves a lot of synthetic homotopy theory: LES of homotopy groups, Hopf fibration, Freudenthal suspension theorem, Blakers-Massey, Z-cohomology (with graded commutative ring structure), Gysin sequence, the Hopf invariant, Whitehead product... Most of this was written by Axel under my supervision, but some results are due to other contributors to the library, in particular Loïc Pujet (3x3 lemma for pushouts, total space of Hopf fibration), KANG Rongji (Blakers-Massey), Evan Cavallo (Freudenthal and lots of clever cubical tricks).
>>
>> Our proof also deviates from the one in Guillaume's thesis in two major ways:
>>
>> 1. We found a direct encode-decode proof of a special case of corollary 3.2.3 and proposition 3.2.11 which is needed for π₄(S³). This allows us to completely avoid the use of the James construction of Section 3 in the thesis (shortening the pen-and-paper proof by ~15 pages), but the price we pay is a less general final result.
>>
>> 2. With Guillaume we have developed a new approach to Z-cohomology in HoTT, in particular to the cup product and cohomology ring (see https://drops.dagstuhl.de/opus/volltexte/2022/15731/). This allows us to give fairly direct construction of the graded commutative ring H*(X;Z), completely avoiding the smash product which has proved very hard to work with formally (and also informally on pen-and-paper as can be seen by the remark in Guillaume's thesis on page 90 just above prop. 4.1.2). This simplification allows us to skip Section 4 of the thesis as well, shortening the pen-and-paper proof by another ~15 pages. This then leads to various further simplifications in Section 5 (Cohomology) and 6 (Gysin sequence).
>>
>> With these mathematical simplifications the proof got a lot more formalization friendly, allowing us to establish an equivalence of groups by a mix of formal proof and computer computations. In particular, Cubical Agda makes it possible to discharge several small steps in the proof involving univalence and HITs purely by computation. This even reduces some gnarly path algebra in the Book HoTT pen-and-paper proof to "refl". Regardless of this, we have not been able to reduce the whole proof to a computation as originally conjectured by Guillaume. However, if someone would be able to do this and compute that the Brunerie number is indeed 2 purely by computer computation there would still be the question what this has to do with π₄(S³). Establishing this connection formally would then most likely involve formalizing (large) parts of what we have managed to do here. Furthermore, having a lot of general theory formalized will enable us to prove more results quite easily which would not be possible from just having a very optimized computation of a specific result.
>>
>> Best regards,
>> Anders and Axel
>>
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  reply	other threads:[~2022-02-09  4:44 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2022-02-08 20:19 Anders Mortberg
2022-02-08 23:24 ` Joyal, André
2022-02-09  3:09   ` Steve Awodey
2022-02-09  4:16     ` Noah Snyder
2022-02-09  4:44       ` 'Urs Schreiber' via Homotopy Type Theory [this message]
2022-02-11 11:29         ` Anders Mortberg

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