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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