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Feb 8, 2022, 3:20:23 PM2/8/22

to Homotopy Type Theory

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

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.

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 AxelFeb 8, 2022, 6:24:37 PM2/8/22

to Anders Mortberg, Homotopy Type Theory

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

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Feb 8, 2022, 10:09:50 PM2/8/22

to "Joyal, André", Anders Mortberg, Homotopy Type Theory

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

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Feb 8, 2022, 11:16:56 PM2/8/22

to Steve Awodey, Joyal, André, Anders Mortberg, Homotopy Type Theory

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

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Feb 8, 2022, 11:44:37 PM2/8/22

to Homotopy Type Theory, Steve Awodey, Joyal, André, Anders Mortberg

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

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(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

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Feb 11, 2022, 6:30:16 AM2/11/22

to Homotopy Type Theory, Steve Awodey, Joyal, André, Urs Schreiber

Thanks a lot to André, Steve and everyone else that wrote to us privately for your kind words and congratulations! It's very nice to be 100% sure that the Brunerie number is indeed 2 after spending far too much time and computer power on trying to compute it :-)

I agree with André that this shows that Cubical Agda is a great tool for developing a lot of homotopy theory, however there are two issues that I should mention:

1. It's unknown if Cubical Agda has a model in spaces (i.e. Kan sSet).

2. It's unknown if any cubical type theory has models in some wide class of infty-categories/topoi.

I find 1 less interesting than 2 as π₄(S³)≅ℤ/2ℤ is a very well established result in spaces (thanks to Noah and Urs for the references!). Furthermore, I expect there to be no problem to translate the proof to cartesian cubical type theory which can then be interpreted in spaces (using the equivariant cartesian model of Awodey-Cavallo-Coquand-Riehl-Sattler). It would of course be a lot of engineering work to change Cubical Agda to be based on cartesian ctt and to rewrite the proof, so I would much rather see some kind of conservativity result relating the two type theories... For problem 2 I really hope someone who knows more about infty-categories can make some progress or for someone to prove a conservativity result relating cubical type theory to HoTT (for which the situation is much clearer thanks to Shulman). I guess I could also mention that the Coquand-Huber-Sattler (https://arxiv.org/abs/1902.06572) results don't apply to our proof even if we would remove all reversals. As we still manage to reduce many goals to computations involving univalence our proof would not go through in a cubical type theory where one drops the computation rules for composition/transp/hcomp.

Despite these problems I'm still very pleased that we managed to formalize the result and develop all of this theory formally. I'm also hopeful that we'll find some solutions to the problems above and that we'll one day have a system which has the good computational properties of cubical type theory combined with a lot of interesting models like HoTT.

Best,

Anders

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