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If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a given manifold; the Poincare conjecture is the special case of $X = S^n$ (there are no topologically fake $S^n$s). I'm interested in this in the case of $X = \Bbb{KP}^n$, where $\Bbb K = \Bbb {R,C,H,O}$ (where in the $\Bbb O$ case we fix $n=2$; there are no other objects which deserve to be named "$\Bbb{OP}^n$".) One reason to find this interesting is that fake projective spaces are precisely the manifolds that have a sphere bundle over them with total space a sphere, with as usual the unfortunate exception that this is not true for $\Bbb{OP}^2$. (I would note that this is different than the algebro-geometric usage of "fake projective plane", where to my understanding what was sought was a classification of compact complex manifolds with the same Hodge diamond as complex projective space.)

Both real and complex fake projective spaces have been classified: both computations are carried out in Wall's book on surgery theory (see sections 14C and 14D for the classifications of fake complex and real projective spaces, respectively), and there are nice, briefer descriptions of the real and complex cases on the Manifold Atlas.

I've had some difficulty in finding a classification of fake quaternionic projective spaces or fake octonionic projective planes. Has this been carried out, and if so, what is a reference?

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    $\begingroup$ $\mathbb{OP}^1=S^8$ $\endgroup$ – David Roberts May 14 '16 at 5:43
  • $\begingroup$ Montgomery and Yang wrote a series of papers about the case of $\mathbb{K}=\mathbb{C}$, but I don't know if they achieved a classification. $\endgroup$ – Ben McKay May 16 '16 at 9:13
  • $\begingroup$ @BenMcKay Wall's book provides a PL classification in that case. It seems Montgomery-Yang were interested in the smooth case (and later the case of not-quite-free actions). Wall cites Brumfiel, “Differentiable $S^1$-actions on homotopy spheres", as what seems to be the authoritative reference; I don't know how up-to-date this is. In general, I'm satisfied with a PL classification, or otherwise I would back up to being unsatisfied with the case of $S^n$... $\endgroup$ – Mike Miller May 17 '16 at 2:22
  • $\begingroup$ @MikeMiller: Do you know of a reference for the statement that "fake projective spaces are precisely the manifolds that have a sphere bundle over them with total space a sphere"? $\endgroup$ – Dan Ramras Jun 19 '17 at 1:34
  • $\begingroup$ @DanRamras I don't know a reference, but I can give you a relatively short proof (assuming 'sphere bundle' is assumed to mean unit sphere bundle of some vector bundle). Feel free to email me. $\endgroup$ – Mike Miller Jun 19 '17 at 1:48
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It seems that fake projective planes (the case $n=2$) have been completely classified, following work of Eells-Kuiper and Kramer-Stolz:

James Eells, Jr. and Nicolaas H. Kuiper, MR 145544 Manifolds which are like projective planes, Inst. Hautes \'Etudes Sci. Publ. Math. (1962), no. 14, 5--46.

Linus Kramer and Stephan Stolz, MR 2355782 A diffeomorphism classification of manifolds which are like projective planes, J. Differential Geom. 77 (2007), no. 2, 177--188.

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  • $\begingroup$ Great! (That second paper is lovely; I haven't had a chance to look in detail at the first.) So we're reduced to the case of the quaternionic projective spaces. $\endgroup$ – Mike Miller May 14 '16 at 14:33
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I would like to add some more references concerning the (smooth) fake quaternionic projective spaces (FQPS). There is a beautiful paper of Hsiang:

Hsiang, Wu-chung A note on free differentiable actions of S1 and S3 on homotopy spheres. Ann. of Math. (2) 83 1966 266–272

where he proves that there are infinitely many FQPS which can be distinguished by their rational Pontryagin classes. Moreover he shows that every FQPS can be obtained by a smooth and free action of $S^3$ on an exotic sphere.

Finally there is a recent preprint where the authors compute some interesting groups concerning smooth structures on $\mathbb H\mathbb P^ n$ for low $n$.

But in general I believe there is not much known and I think it is interesting subject to think about.

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