From: Anders Mortberg <email@example.com>
To: Homotopy Type Theory <firstname.lastname@example.org>
Cc: "Axel Ljungström" <email@example.com>
Subject: [HoTT] The Brunerie number is -2
Date: Mon, 23 May 2022 21:30:13 +0200 [thread overview]
Message-ID: <CAMWCppkF0JTQ8z6sPgLaC1=NZYFQdocCjUamCUDJUwGu179XXw@mail.gmail.com> (raw)
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We're very happy to announce that we have finally managed to compute the
Brunerie number using Cubical Agda... and the result is -2!
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
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
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.
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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next reply other threads:[~2022-05-23 19:30 UTC|newest]
Thread overview: 6+ messages / expand[flat|nested] mbox.gz Atom feed top
2022-05-23 19:30 Anders Mortberg [this message]
2022-05-23 19:38 ` Steve Awodey
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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