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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeMar 25th 2019

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeMar 25th 2019

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 25th 2019
• (edited Mar 25th 2019)

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

$\array{ S^7 &\longrightarrow& B G_2 &\longrightarrow& B Spin(7) \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ S^7 &\longrightarrow& B Spin(7) &\longrightarrow& B Spin(8) }$
• CommentRowNumber4.
• CommentAuthorDavidRoberts
• CommentTimeMar 25th 2019

Intriguing!

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeMar 25th 2019
• (edited Mar 25th 2019)

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?

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeMar 26th 2019

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

• $Spin(6)/SU(3)$

• $Spin(5)/SU(2)$

• CommentRowNumber7.
• CommentAuthorDavidRoberts
• CommentTimeMar 26th 2019

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

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeMar 26th 2019
• (edited Mar 26th 2019)

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

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMar 27th 2019

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…)

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeMar 27th 2019
• (edited Mar 27th 2019)

[ removed ]