The Brunerie number is ±2 - direct proof #741
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aljungstrom
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aljungstrom
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This PR contains a direct proof of the fact that the Brunerie number is ±2 (in fact, with my construction, it's -2). The proof itself is relatively straightforward (see Cubical.Homotopy.Group.Pi4S3.QuickProof -- you may suggest a better name...) but relies on some machinery concerning the equivalence
Sⁿ * Sᵐ ≃ Sⁿ⁺ᵐ. The idea is to formulate π₃(A) asπ*₃(A) := ∥ S¹ * S¹ →∙ A ∥₂(with an explicitly defined multiplication) and analyse the termη₁ : π₃*(S²)defining the Brunerie number as a member of this group. We can then prove directly that it gets mapped to -2 under what is pretty much the usual equivalenceπ₃(S²) ≅ ℤ.We get a bunch of terms corresponding to η on the way:
η₁ : π₃*(S²), η₂ : π₃*(S¹ * S¹), η₃ : π₃*(S³) , η₄ : π₃(S³)which we can try to send to ℤ. This gives equivalent, but hopefully simpler, computations of the Brunerie number (I haven't tried this yet).