Re: [HoTT] The Brunerie number is -2

[Nicolai Kraus <nicolai.kraus@gmail.com>]

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Congratulations! It's great that this number finally computes in practice
and not just in theory, after all these years. :-)
And it's impressive how short the new proof is! But this still doesn't mean
that Cubical Agda passes the test that Guillaume formulates in Appendix B
of his thesis, right? Because this test refers to the Brunerie number β (in
the Summary.agda file you linked), and not to β'.
In any case, that's a fantastic result!
Best,
Nicolai







On Mon, May 23, 2022 at 8:30 PM Anders Mortberg <andersmortberg@gmail.com>
wrote:

> We're very happy to announce that we have finally managed to compute the
> Brunerie number using Cubical Agda... and the result is -2!
>
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/Summary.agda#L129
>
> The computation was made possible by a new direct synthetic proof that
> pi_4(S^3) = Z/2Z by Axel Ljungström. This new proof involves a series of
> new Brunerie numbers (i.e. numbers n : Z such that pi_4(S^3) = Z/nZ) and we
> got the one called β' in the file above to reduce to -2 in just a few
> seconds. With some work we then managed to prove that pi_4(S^3) = Z / β' Z,
> leading to a proof by normalization of the number as conjectured in
> Brunerie's thesis.
>
> Axel's new proof is very direct and completely avoids chapters 4-6 in
> Brunerie's thesis (so no cohomology theory!), but it relies on chapters 1-3
> to define the number. It also does not rely on any special features of
> cubical type theory and should be possible to formalize also in systems
> based on Book HoTT. For a proof sketch as well as the formalization of the
> new proof in just ~700 lines (not counting what is needed from chapters
> 1-3) see:
>
>
> https://github.com/agda/cubical/blob/master/Cubical/Homotopy/Group/Pi4S3/DirectProof.agda
>
> So to summarize we now have both a new direct HoTT proof, not relying on
> cubical computations, as well as a cubical proof by computation.
>
> Univalent regards,
> Anders and Axel
>
> PS: the minus sign is actually not very significant and we can get +2 by
> slightly modifying β', but it's quite funny that we ended up getting -2
> when we finally got a definition which terminates!
>
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