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added pointer to
added this statement:
Consider the canonical action of Spin(7) on the unit sphere in $\mathbb{R}^8$ (the 7-sphere),
This action is is transitive;
the stabilizer group of any point on $S^7$ is G2;
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
$Spin(7)/G2 \;\simeq\; S^7 \,.$Hm, so it must be that we have this situation:
$\array{ S^7 &\longrightarrow& B G_2 &\longrightarrow& B Spin(7) \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ S^7 &\longrightarrow& B Spin(7) &\longrightarrow& B Spin(8) }$Intriguing!
What further realizations of 7-spheres (or 4-spheres) as coset spaces do we have?
So far I know that $S^7 \simeq \cdots$
$\cdots Spin(8)/Spin(7)$ (clear)
$\cdots Spin(7)/G_2$ (by what we just said)
$\cdots Sp(2)/(Sp(1) \times Sp(1))$ (the Gromoll-Meyer sphere)
Anything else? I’d like to see more of the exceptional Lie groups show up – any chance?
John Baez wrote a post years ago about 4 different ways to build the 7-sphere as a homogeneous space, including
$Spin(6)/SU(3)$
$Spin(5)/SU(2)$
So I guess that last one is $Sp(2)/SU(2)$, which is pleasing.
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…)
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