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Shouldn’t $S^15 \simeq_{top} Spin(9)/Spin(7)$ count as ’exceptional’? It’s not in the list as it stands.
There’s also the family $S^{4 n -1} \simeq Sp(n)/Sp(n-1)$. Is that a diffeomorphism?
But then $S^7 \simeq Spin(5)/SU(2)$ wouldn’t be exceptional, as in the list, since it’s just $Sp(2)/Sp(1)$.
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.
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.
Sure, that’s a good point. Thanks for adding.
added a pointer, to Borel-Serre 53, 17.1, though there must be more canonical references
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?
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.
Are coset spaces that arise from “exceptional isomorphisms” really exceptional?
Edit: By this I mean don’t they just arise from the “unexceptional” families?
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).
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.
somebody should add exotic examples
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
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 $E^{11}$ on slide 23 of 38. Euclidean space?
Need to read in more detail…
Used unicode subscripts for indices of exceptional Lie groups. See discussion here.
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