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nLab > Latest Changes: coset space structure on n-spheres -- table

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1 to 18 of 18

  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 19th 2019

    summary table, to be !includeed into relevant entries, for purposes of cross-linking

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019

    Shouldn’t S 15≃ topSpin(9)/Spin(7)S^15 \simeq_{top} Spin(9)/Spin(7) count as ’exceptional’? It’s not in the list as it stands.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019

    There’s also the family S 4n−1≃Sp(n)/Sp(n−1)S^{4 n -1} \simeq Sp(n)/Sp(n-1). Is that a diffeomorphism?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019

    But then S 7≃Spin(5)/SU(2)S^7 \simeq Spin(5)/SU(2) wouldn’t be exceptional, as in the list, since it’s just Sp(2)/Sp(1)Sp(2)/Sp(1).

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 24th 2019

    Yes, there are more “exceptional” spheres, if one goes to higher dimensions. The table can be expanded.

    Not sure why you wouldn’t count that S 7S^7 as “exceptional”. Of course there is some subjectivity involved in that term.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019
    • (edited Apr 24th 2019)

    Not sure why you wouldn’t count that S 7S^7 as “exceptional”.

    In #3 I was suggesting that the family S 4n−1≃Sp(n)/Sp(n−1)S^{4 n -1} \simeq Sp(n)/Sp(n-1) be added to the two ’standard’ entries. Then that case Spin(5)/SU(2)Spin(5)/SU(2) is just a member of the family.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019

    Added in the Sp(n)/Sp(n−1)Sp(n)/Sp(n-1) family.

    diff, v2, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2019

    Sure, that’s a good point. Thanks for adding.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2019

    added a pointer, to Borel-Serre 53, 17.1, though there must be more canonical references

    diff, v3, current

    • CommentRowNumber10.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2019
    • (edited Apr 25th 2019)

    Is Sp(n)/Sp(n−1)Sp(n)/Sp(n-1) a diff or top equivalence? If diff, then that should apply to Spin(5)/SU(2)Spin(5)/SU(2), no?

    But then Spin(6)/SU(3)Spin(6)/SU(3) is just an instance of SU(n)/SU(n−1)SU(n)/SU(n-1) and we have the former as top and the latter as diff.

    Or is it that the exceptional isomorphisms are only topological?

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2019

    These should in fact all be diffeomorphisms so far (just not isometries, whence “squashed”), we haven’t started listing exotic examples yet.

    I have edited accordingly. Also, I expanded the line for Spin(5)/SU(2)Spin(5)/SU(2) along the lines you have been suggesting.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2019

    Have added S 15≃ diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7), since this doesn’t fit any family.

    diff, v4, current

    • CommentRowNumber13.
    • CommentAuthorAli Caglayan
    • CommentTimeApr 25th 2019
    • (edited Apr 25th 2019)

    Are coset spaces that arise from “exceptional isomorphisms” really exceptional?

    Edit: By this I mean don’t they just arise from the “unexceptional” families?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2019

    David, thanks for further expanding the list.

    Next, somebody should add exotic examples, such as the Gromoll-Meyer sphere…

    It is striking that Milnor’s construction of exotic 7-spheres finds them as boundaries of 8-manifolds in exactly the way M2-branes appear in M-theory on 8-manifolds (as remarked here).

    • CommentRowNumber15.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2019
    • (edited Apr 25th 2019)

    I was just wondering what element of ℤ/28ℤ\mathbb{Z}/28\mathbb{Z} is the Gromoll-Meyer sphere. Slide 6 here suggests it’s a generator.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2019

    somebody should add exotic examples

    From here

    Recently, it was shown that Σ 7\Sigma^7 is actually the only exotic sphere that can be modeled by a biquotient of a compact Lie group

    Re #15, it says also

    by choosing two local trivializations of this bundle properly, Σ 7\Sigma^7 is identified with the Milnor sphere Σ 2,−1 7\Sigma^7_{2,-1}, which is a generator of the group of homotopy spheres

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeApr 25th 2019

    Slide 6 here suggests

    These are interesting slides! Need to think about this.

    Are all squares with the top morphism dashed meant to be pullback squares?

    What is E 11E^{11} on slide 23 of 38. Euclidean space?

    Need to read in more detail…

  18. Used unicode subscripts for indices of exceptional Lie groups. See discussion here.

    diff, v5, current

1 to 18 of 18

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