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

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  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 S15≃topSpin(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 S4n−1≃Sp(n)/Sp(n−1). Is that a diffeomorphism?

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 24th 2019

    But then S7≃Spin(5)/SU(2) wouldn’t be exceptional, as in the list, since it’s just 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 S7 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 S7 as “exceptional”.

    In #3 I was suggesting that the family S4n−1≃Sp(n)/Sp(n−1) be added to the two ’standard’ entries. Then that case 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) 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) a diff or top equivalence? If diff, then that should apply to Spin(5)/SU(2), no?

    But then Spin(6)/SU(3) is just an instance of 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) along the lines you have been suggesting.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 25th 2019

    Have added S15≃diffSpin(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 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 Σ7 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, Σ7 is identified with the Milnor sphere Σ72,−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 E11 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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