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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexfor completeness, to go with the other entries in _[[coset space structure on n-spheres -- table]]_ <a href="https://ncatlab.org/nlab/revision/Spin%287%29%2FG2+is+the+7-sphere/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29%2FG2+is+the+7-sphere">current</a>

   for completeness, to go with the other entries in coset space structure on n-spheres – table

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorSamuel Adrian Antz
   • CommentTimeJul 17th 2024
   • PermaLink
   Author: Samuel Adrian Antz
   Format: MarkdownItexUsed unicode subscripts for indices of [[exceptional Lie groups]] including title and links. When not linked, usual formulas are used. See discussion [here](https://nforum.ncatlab.org/discussion/18215/title-convention). 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.) <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29%2FG%E2%82%82+is+the+7-sphere/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29%2FG%E2%82%82+is+the+7-sphere/3">v3</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29%2FG%E2%82%82+is+the+7-sphere">current</a>

   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.)

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTimeSep 19th 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexHere's something I find puzzling. In [Stasheff 63](https://doi.org/10.2307/1993608), it is shown that even though the seven-sphere $S^7$ is a [[H-space]], it does not have a homotopy-associative product. The binary operation that makes $S^7$ a H-space is essentially the product of unit octonions. Now, in [Albuquerque & Majid 98](https://arxiv.org/abs/math/9802116), it is explained that the octonions are essentially obtained by starting with the real group algebra $\mathbb{R}[\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 ]$ 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](https://arxiv.org/abs/math/0105155) that one has a (braided) fusion category of $\mathbb{Z}_2 ^3$-graded vector spaces with nontrivial associator. So based on this I thought $S^7$ 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 $Spin(7)//G_2$ that is a (braided) Lie 2-group?

   Here’s something I find puzzling. In Stasheff 63, it is shown that even though the seven-sphere $S^7$ is a H-space, it does not have a homotopy-associative product. The binary operation that makes $S^7$ 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 $\mathbb{R}[\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2 ]$ 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 $\mathbb{Z}_2 ^3$-graded vector spaces with nontrivial associator. So based on this I thought $S^7$ 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 $Spin(7)//G_2$ that is a (braided) Lie 2-group?