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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to * [[Veeravalli Varadarajan]], _Spin(7)-subgroups of SO(8) and Spin(8)_, Expositiones Mathematicae, 19 (2001): 163-177 ([pdf](https://core.ac.uk/download/pdf/81114499.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29">current</a>

   added pointer to

   • Veeravalli Varadarajan, Spin(7)-subgroups of SO(8) and Spin(8), Expositiones Mathematicae, 19 (2001): 163-177 (pdf)

   diff, v4, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this statement: *** Consider the canonical [[action]] of [[Spin(7)]] on the [[unit sphere]] in $\mathbb{R}^8$ (the [[7-sphere]]), 1. This action is is [[transitive action|transitive]]; 1. the [[stabilizer group]] of any point on $S^7$ is [[G2]]; 1. all [[G2]]-subgroups of [[Spin(7)]] arise this way, and are all [[conjugate subgroup|conjugate]] to each other. Hence the [[coset space]] of [[Spin(7)]] by [[G2]] is the [[7-sphere]] $$ Spin(7)/G2 \;\simeq\; S^7 \,. $$ *** will copy this also to _[[G2]]_ and _[[7-sphere]]_ <a href="https://ncatlab.org/nlab/revision/diff/Spin%287%29/4">diff</a>, <a href="https://ncatlab.org/nlab/revision/Spin%287%29/4">v4</a>, <a href="https://ncatlab.org/nlab/show/Spin%287%29">current</a>

   added this statement:

   ---

   Consider the canonical action of Spin(7) on the unit sphere in $\mathbb{R}^8$ (the 7-sphere),

   1. This action is is transitive;

   2. the stabilizer group of any point on $S^7$ is G2;

   3. all G2-subgroups of Spin(7) arise this way, and are all conjugate to each other.

   Hence the coset space of Spin(7) by G2 is the 7-sphere

   $$Spin(7)/G2 \;\simeq\; S^7 \,.$$

   ---

   will copy this also to G2 and 7-sphere

   diff, v4, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2019
   • (edited Mar 25th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHm, so it must be that we have this situation: $$ \array{ S^7 &\longrightarrow& B G_2 &\longrightarrow& B Spin(7) \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ S^7 &\longrightarrow& B Spin(7) &\longrightarrow& B Spin(8) } $$

   Hm, so it must be that we have this situation:

   $$\array{ S^7 &\longrightarrow& B G_2 &\longrightarrow& B Spin(7) \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ S^7 &\longrightarrow& B Spin(7) &\longrightarrow& B Spin(8) }$$
4. • CommentRowNumber4.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 25th 2019
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexIntriguing!

   Intriguing!

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 25th 2019
   • (edited Mar 25th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWhat further realizations of 7-spheres (or 4-spheres) as coset spaces do we have? So far I know that $S^7 \simeq \cdots$ 1. $\cdots Spin(8)/Spin(7)$ (clear) 1. $\cdots Spin(7)/G_2$ (by what we just said) 1. $\cdots Sp(2)/(Sp(1) \times Sp(1))$ (the [[Gromoll-Meyer sphere]]) Anything else? I'd like to see more of the exceptional Lie groups show up -- any chance?

   What further realizations of 7-spheres (or 4-spheres) as coset spaces do we have?

   So far I know that $S^7 \simeq \cdots$

   1. $\cdots Spin(8)/Spin(7)$ (clear)

   2. $\cdots Spin(7)/G_2$ (by what we just said)

   3. $\cdots Sp(2)/(Sp(1) \times Sp(1))$ (the Gromoll-Meyer sphere)

   Anything else? I’d like to see more of the exceptional Lie groups show up – any chance?

6. • CommentRowNumber6.
   • CommentAuthorDavid_Corfield
   • CommentTimeMar 26th 2019
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexJohn Baez wrote a [post](https://golem.ph.utexas.edu/category/2007/09/rotations_in_the_7th_dimension.html) years ago about 4 different ways to build the 7-sphere as a homogeneous space, including * $ Spin(6)/SU(3)$ * $Spin(5)/SU(2)$

   John Baez wrote a post years ago about 4 different ways to build the 7-sphere as a homogeneous space, including

   • $Spin(6)/SU(3)$

   • $Spin(5)/SU(2)$

7. • CommentRowNumber7.
   • CommentAuthorDavidRoberts
   • CommentTimeMar 26th 2019
   • PermaLink
   Author: DavidRoberts
   Format: MarkdownItexSo I guess that last one is $Sp(2)/SU(2)$, which is pleasing.

   So I guess that last one is $Sp(2)/SU(2)$, which is pleasing.

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeMar 26th 2019
   • (edited Mar 26th 2019)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI only now realize that my reply to the above got sent to the wrong thread ([here](https://nforum.ncatlab.org/discussion/6721/7sphere/?Focus=76867#Comment_76867)), where I said: Thanks! Have added these [here](https://ncatlab.org/nlab/show/7-sphere#CosetSpaceRealization) <a href="https://ncatlab.org/nlab/revision/diff/7-sphere/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/7-sphere/20">v20</a>, <a href="https://ncatlab.org/nlab/show/7-sphere">current</a>

   I only now realize that my reply to the above got sent to the wrong thread (here), where I said:

   Thanks! Have added these here

   diff, v20, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2019
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo the diagram in #3 is just the last in a pasting array of similar diagrams, the de-homotopified version of which is a diagram of subgroup intersections in $Spin(8)$ which I just typed out [here](https://ncatlab.org/nlab/show/SO8#SubgroupLattice) (in `xymatrix`, so it doesn't display here...)

   So the diagram in #3 is just the last in a pasting array of similar diagrams, the de-homotopified version of which is a diagram of subgroup intersections in $Spin(8)$ which I just typed out here

   (in xymatrix, so it doesn’t display here…)

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 27th 2019
    • (edited Mar 27th 2019)
    • PermaLink
    Author: Urs
    Format: MarkdownItex[ removed ]

    [ removed ]