Discussion of Homotopy Type Theory and Univalent Foundations
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From: Steve Awodey <steveawodey@gmail.com>
To: Anders Mortberg <andersmortberg@gmail.com>
Cc: "Homotopy Type Theory" <homotopytypetheory@googlegroups.com>,
	"Axel Ljungström" <axel.ljungstrom@math.su.se>
Subject: Re: [HoTT] The Brunerie number is -2
Date: Mon, 23 May 2022 21:38:55 +0200	[thread overview]
Message-ID: <3D8F3A1E-0E9A-47A1-A3B7-CB05B03B802D@gmail.com> (raw)
In-Reply-To: <CAMWCppkF0JTQ8z6sPgLaC1=NZYFQdocCjUamCUDJUwGu179XXw@mail.gmail.com>

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Congratulations to Team A^2!
Great work - and a real milestone. 
Best wishes,
Steve 

> On May 23, 2022, at 21:30, 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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  reply	other threads:[~2022-05-23 19:38 UTC|newest]

Thread overview: 6+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2022-05-23 19:30 Anders Mortberg
2022-05-23 19:38 ` Steve Awodey [this message]
2022-05-23 20:22 ` Nicolai Kraus
2022-05-23 20:59   ` Anders Mortberg
2022-05-24  9:46     ` Anders Mörtberg
2022-05-24  9:49       ` Anders Mörtberg

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