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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 19th 2019

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

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeApr 24th 2019

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

• CommentRowNumber3.
• CommentAuthorDavid_Corfield
• CommentTimeApr 24th 2019

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

• CommentRowNumber4.
• CommentAuthorDavid_Corfield
• CommentTimeApr 24th 2019

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

• 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 $S^7$ 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 $S^7$ 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.

• CommentRowNumber7.
• CommentAuthorDavid_Corfield
• CommentTimeApr 24th 2019

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

• 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

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

• 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 $\mathbb{Z}/28\mathbb{Z}$ is the Gromoll-Meyer sphere. Slide 6 here suggests it’s a generator.

• CommentRowNumber16.
• CommentAuthorDavid_Corfield
• CommentTimeApr 25th 2019

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

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeApr 25th 2019

Slide 6 here suggests

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