From: Anders Mortberg <andersmortberg@gmail.com> To: Homotopy Type Theory <homotopytypetheory@googlegroups.com> Cc: "Axel Ljungström" <axel.ljungstrom@math.su.se> 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) [-- Attachment #1: Type: text/plain, Size: 2037 bytes --] 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! -- You received this message because you are subscribed to the Google Groups "Homotopy Type Theory" group. To unsubscribe from this group and stop receiving emails from it, send an email to HomotopyTypeTheory+unsubscribe@googlegroups.com. To view this discussion on the web visit https://groups.google.com/d/msgid/HomotopyTypeTheory/CAMWCppkF0JTQ8z6sPgLaC1%3DNZYFQdocCjUamCUDJUwGu179XXw%40mail.gmail.com. [-- Attachment #2: Type: text/html, Size: 2928 bytes --]

next reply other threads:[~2022-05-23 19:30 UTC|newest]Thread overview:6+ messages / expand[flat|nested] mbox.gz Atom feed top2022-05-23 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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