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for completeness, to go with the other entries in coset space structure on n-spheres – table
Used unicode subscripts for indices of exceptional Lie groups including title and links. When not linked, usual formulas are used. See discussion here. Links will be re-checked after all titles have been changed. (Removed two redirects for “Spin(7)/G2 is the 7-sphere” from the top and added one for “Spin(7)/G2 is the 7-sphere” at the bottom of the page.)
Here’s something I find puzzling. In Stasheff 63, it is shown that even though the seven-sphere S7 is a H-space, it does not have a homotopy-associative product. The binary operation that makes S7 a H-space is essentially the product of unit octonions. Now, in Albuquerque & Majid 98, it is explained that the octonions are essentially obtained by starting with the real group algebra ℝ[ℤ2×ℤ2×ℤ2] and twisting the multiplication by some 2-cochain. Since this cochain is not a cocycle, the resulting multiplication is not associative, but the nontrivial associator will satisfy a 3-cocycle condition. This is essentially what leads to the observation in p.9 in Baez 01 that one has a (braided) fusion category of ℤ32-graded vector spaces with nontrivial associator. So based on this I thought S7 would admit a homotopy associative multiplication, but this is not the case. Seems something is lost here. Is this suggesting that it is the stack Spin(7)//G2 that is a (braided) Lie 2-group?
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