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…

]]>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

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.

]]>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).

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

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

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

]]>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.

]]>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?

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

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

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

]]>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.

]]>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.

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

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

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

]]>summary table, to be `!include`

ed into relevant entries, for purposes of cross-linking