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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019

    added pointer to

    diff, v4, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 25th 2019

    added this statement:


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

    1. This action is is transitive;

    2. the stabilizer group of any point on S 7S^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)/G2S 7. Spin(7)/G2 \;\simeq\; S^7 \,.

    will copy this also to G2 and 7-sphere

    diff, v4, current

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

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

    S 7 BG 2 BSpin(7) (pb) S 7 BSpin(7) BSpin(8) \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 7S^7 \simeq \cdots

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

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

    3. Sp(2)/(Sp(1)×Sp(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?

    • 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(6)/SU(3)

    • Spin(5)/SU(2)Spin(5)/SU(2)

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 26th 2019

    So I guess that last one is Sp(2)/SU(2)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:

    Thanks! Have added these here

    diff, v20, current

    • 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)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 ]

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