nForum - Discussion Feed (coset space structure on n-spheres -- table) 2023-06-02T14:37:47+00:00 https://nforum.ncatlab.org/ Lussumo Vanilla & Feed Publisher Urs comments on "coset space structure on n-spheres -- table" (77573) https://nforum.ncatlab.org/discussion/9833/?Focus=77573#Comment_77573 2019-04-25T16:39:17+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ 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. ...

Slide 6 here suggests

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

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

Need to read in more detail…

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David_Corfield comments on "coset space structure on n-spheres -- table" (77572) https://nforum.ncatlab.org/discussion/9833/?Focus=77572#Comment_77572 2019-04-25T16:35:49+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ somebody should add exotic examples From here Recently, it was shown that &Sigma; 7\Sigma^7 is actually the only exotic sphere that can be modeled by a biquotient of a compact Lie ...

From here

Recently, it was shown that $\Sigma^7$ 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, $\Sigma^7$ is identified with the Milnor sphere $\Sigma^7_{2,-1}$, which is a generator of the group of homotopy spheres

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David_Corfield comments on "coset space structure on n-spheres -- table" (77571) https://nforum.ncatlab.org/discussion/9833/?Focus=77571#Comment_77571 2019-04-25T16:26:06+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ I was just wondering what element of &Zopf;/28&Zopf;\mathbb{Z}/28\mathbb{Z} is the Gromoll-Meyer sphere. Slide 6 here suggests it’s a generator.

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

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Urs comments on "coset space structure on n-spheres -- table" (77570) https://nforum.ncatlab.org/discussion/9833/?Focus=77570#Comment_77570 2019-04-25T15:43:31+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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).

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Ali Caglayan comments on "coset space structure on n-spheres -- table" (77568) https://nforum.ncatlab.org/discussion/9833/?Focus=77568#Comment_77568 2019-04-25T14:27:45+00:00 2023-06-02T14:37:46+00:00 Ali Caglayan https://nforum.ncatlab.org/account/1731/ Are coset spaces that arise from “exceptional isomorphisms” really exceptional? Edit: By this I mean don’t they just arise from the “unexceptional” families?

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

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

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David_Corfield comments on "coset space structure on n-spheres -- table" (77566) https://nforum.ncatlab.org/discussion/9833/?Focus=77566#Comment_77566 2019-04-25T14:22:40+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Have added S 15&simeq; diffSpin(9)/Spin(7)S^15 \simeq_{diff} Spin(9)/Spin(7), since this doesn’t fit any family. diff, v4, current

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

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Urs comments on "coset space structure on n-spheres -- table" (77563) https://nforum.ncatlab.org/discussion/9833/?Focus=77563#Comment_77563 2019-04-25T13:57:14+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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.

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David_Corfield comments on "coset space structure on n-spheres -- table" (77562) https://nforum.ncatlab.org/discussion/9833/?Focus=77562#Comment_77562 2019-04-25T13:50:13+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Is Sp(n)/Sp(n&minus;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 ...

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?

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Urs comments on "coset space structure on n-spheres -- table" (77561) https://nforum.ncatlab.org/discussion/9833/?Focus=77561#Comment_77561 2019-04-25T13:37:55+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ added a pointer, to Borel-Serre 53, 17.1, though there must be more canonical references diff, v3, current

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

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Urs comments on "coset space structure on n-spheres -- table" (77558) https://nforum.ncatlab.org/discussion/9833/?Focus=77558#Comment_77558 2019-04-25T05:10:14+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ Sure, that’s a good point. Thanks for adding.

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

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David_Corfield comments on "coset space structure on n-spheres -- table" (77552) https://nforum.ncatlab.org/discussion/9833/?Focus=77552#Comment_77552 2019-04-24T16:06:53+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Added in the Sp(n)/Sp(n&minus;1)Sp(n)/Sp(n-1) family. diff, v2, current

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

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David_Corfield comments on "coset space structure on n-spheres -- table" (77550) https://nforum.ncatlab.org/discussion/9833/?Focus=77550#Comment_77550 2019-04-24T15:53:03+00:00 2023-06-02T14:37:46+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Not sure why you wouldn’t count that S 7S^7 as “exceptional”. In #3 I was suggesting that the family S 4n&minus;1&simeq;Sp(n)/Sp(n&minus;1)S^{4 n -1} \simeq Sp(n)/Sp(n-1) be added ...

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

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

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Urs comments on "coset space structure on n-spheres -- table" (77549) https://nforum.ncatlab.org/discussion/9833/?Focus=77549#Comment_77549 2019-04-24T15:39:59+00:00 2023-06-02T14:37:46+00:00 Urs https://nforum.ncatlab.org/account/4/ 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 ...

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 $S^7$ as “exceptional”. Of course there is some subjectivity involved in that term.

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David_Corfield comments on "coset space structure on n-spheres -- table" (77548) https://nforum.ncatlab.org/discussion/9833/?Focus=77548#Comment_77548 2019-04-24T14:09:37+00:00 2023-06-02T14:37:47+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ But then S 7&simeq;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).

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

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David_Corfield comments on "coset space structure on n-spheres -- table" (77547) https://nforum.ncatlab.org/discussion/9833/?Focus=77547#Comment_77547 2019-04-24T13:40:01+00:00 2023-06-02T14:37:47+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ There’s also the family S 4n&minus;1&simeq;Sp(n)/Sp(n&minus;1)S^{4 n -1} \simeq Sp(n)/Sp(n-1). Is that a diffeomorphism?

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

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David_Corfield comments on "coset space structure on n-spheres -- table" (77546) https://nforum.ncatlab.org/discussion/9833/?Focus=77546#Comment_77546 2019-04-24T13:27:22+00:00 2023-06-02T14:37:47+00:00 David_Corfield https://nforum.ncatlab.org/account/20/ Shouldn’t S 15&simeq; topSpin(9)/Spin(7)S^15 \simeq_{top} Spin(9)/Spin(7) count as ’exceptional’? It’s not in the list as it stands.

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

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Urs comments on "coset space structure on n-spheres -- table" (77470) https://nforum.ncatlab.org/discussion/9833/?Focus=77470#Comment_77470 2019-04-19T20:14:20+00:00 2023-06-02T14:37:47+00:00 Urs https://nforum.ncatlab.org/account/4/ summary table, to be !includeed into relevant entries, for purposes of cross-linking v1, current

summary table, to be `!include`ed into relevant entries, for purposes of cross-linking

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