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Let $M^n$ be a smooth manifold whose universal cover is homeomorphic $\mathbb{S}^n$, are there examples where $M^n$ is not homeomorphic to a space form ?

The answer may vary if you replace homeomorphic by diffeomorphic, and I'm also interested in this question under this restriction.

I came to this question while reading surveys about sphere theorems, where the non simply-connected case is harder to get (while you already know that the universal cover is a sphere, you need more work to show it is actually a space-form).

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  • $\begingroup$ Thanks to both Igors for your answers. I'm not familiar with these topics, so I'll have to dig a bit into these papers to understand. Does there exist a more elementary construction ? $\endgroup$
    – Thomas Richard
    Commented Sep 18, 2012 at 12:58
  • $\begingroup$ The reference I suggest is a book, so this is probably as gentle as you can hope for. This was one of the guiding problems in topology for a couple of decades, so I don't think there is a really easy way of getting there. $\endgroup$
    – Igor Rivin
    Commented Sep 18, 2012 at 13:13
  • $\begingroup$ Interestingly, the book Igor Rivin links to defines a space form to be "a manifold whose universal cover is a sphere" (on the first page of the introduction). I guess that here we mean "manifold which is the orbit space of a free action on a sphere". $\endgroup$
    – Mark Grant
    Commented Sep 18, 2012 at 13:34
  • $\begingroup$ There are other examples. You might like fake $RP^5$'s of nonnegative curvature of Grove-Ziller, see Theorem G in math.upenn.edu/~wziller/papers/groveziller.ps. Of course, the existence of fake $RP^5$'s was known to topologists long ago, but the above paper gives a geometric construction. Is that what you seek? $\endgroup$
    – Igor Belegradek
    Commented Sep 18, 2012 at 13:44
  • $\begingroup$ Are you interested when the universal cover of $M$ is an exotic sphere (and of course $M$ is smooth)? $\endgroup$
    – Ian Agol
    Commented Sep 18, 2012 at 14:34

3 Answers 3

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There are lots of fake lens spaces, and fake spherical space forms (search on these keywords). In particular, a construction of fake lens spaces is in chapter 12 of Milnor's "Whitehead torsion". Here are details: start with any lens space $L$ with fundamental group of order different from 2,3,4, 6, so that its Whitehead group is infinite. Then there are infinitely many distict manifolds that are h-cobordant to $L$. On the other hand, Corollary 12.13 implies that any h-cobordism between lens spaces is a product. (Of course, I assume dimension $\ge 5$ here).

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See Davis-Milgram

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The geometrization theorem implies that 3-dimensional manifolds covered by $S^3$ are diffeomorphic to space forms.

There are examples of Cappell-Shaneson and Fintushel-Stern of fake $\mathbb{RP}^4$s (in the Fintushel-Stern case, they actually show there is a smooth exotic free involution of $S^4$).

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