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I like to explore and ask for proper references for the relations between generalized projective spaces, spheres, and exotic spheres:

  1. The real projective space $\mathbb{RP}^1 \simeq S^1,$ is homeomorphic and diffeomorphic to 1-sphere $S^1.$

  2. The complex projective space $\mathbb{CP}^1 \simeq S^2,$ is homeomorphic and diffeomorphic to 2-sphere $S^2.$

  3. The quaternion projective space $\mathbb{HP}^1 \simeq S^4,$ is homeomorphic and diffeomorphic to 4-sphere $S^4.$

I am looking for general constructions of generalized projective spaces, and their relations to spheres, and exotic spheres. For example, we can consider generalized projective spaces homeomorphic to $S^8,S^{16},\dots, S^{2^n},$ etc., for $n\in \mathbb{N}$(yes ?).

  • Are they necessarily diffeomorphic to the standard spheres?

  • Are there generalized projective spaces of exotic spheres?

  • Can one consider the quotient manifold of generalized projective spaces to construct spheres, and/or exotic spheres?

(e.g. The 8-dimensional $\mathbb{HP}^{2}$ has a circle action, by the group of complex scalars of absolute value 1 acting on the other side (so on the right, instead of on the left action). Therefore we can obtain the quotient manifold $\mathbb{HP}^{2}/\mathrm{U}(1)$ with U(1) for the circle group. It has been shown that this quotient is the 7-sphere, a result of Arnold (1996), also later by Witten and Atiyah.)

Thank you in advance!

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    $\begingroup$ What is a generalized projective space? I don't know of anything continuing the sequence RP^1, CP^1, HP^1, other than the octonionic projective line OP^1 (which is indeed homeomorphic and diffeomorphic to S^8). What's supposed to "come next"? (Maybe this is the content of the question.) $\endgroup$
    – skd
    Commented Jul 9, 2018 at 22:50
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    $\begingroup$ These "algebras" have something to do wtih hopf invariants, Adams and then later Atiyah proved that there are four maps with hopf invariant 1. These correspond to $\Bbb {R, C, H, O}$. Infact this is how Atiyah used topological K-theory to prove that these are the only normed division "algebras" (octionions aren't quite associative but close enough). The point is if there was another "projective space" the hopf invariant would have to be wrong. I am guessing this argument as I am no where near an expert. $\endgroup$
    – Ali Caglayan
    Commented Jul 10, 2018 at 11:08
  • $\begingroup$ @AliCaglayan you don't need Hopf invariant one to construct projective lines as CW-complexes; that result only prohibits the existence of anything past OP^2. Note however that OP^3 doesn't exist, but this is not because of Hopf invariant one. $\endgroup$
    – skd
    Commented Jul 10, 2018 at 12:48
  • $\begingroup$ @skd I was not thinking in that direction, I was trying to explain why you can't just cayley-dickson octionions and make a projective line. $\endgroup$
    – Ali Caglayan
    Commented Jul 10, 2018 at 15:53

1 Answer 1

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While I think it is difficult to give a solid answer to this question, you may perhaps be interested in Projective planes, Severi varieties and spheres from 2002, by Atiyah and Berndt.

For example they show that $\mathbb{OP}^2 / Sp(1) \simeq S^{13}$ naturally follows the $\mathbb{HP}^2/ U(1) \simeq S^7$ diffeomorphism that you mention, and they give a fairly-extensive discussion of the context.

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