Re: [HoTT] Formalization of π₄(S³)≅ℤ/2ℤ in Cubical Agda completed

[Anders Mortberg <andersmortberg@gmail.com>]

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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


On Wed, Feb 9, 2022 at 5:44 AM Urs Schreiber <urs.schreiber@googlemail.com>
wrote:

> 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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