Suppose that the smooth manifold $ M $ has the n-sphere for its universal cover (in the topological sense). Does there exist a Riemannian metric on $ M $ (not necessarily compatible with the covering map) for which all sectional curvatures of $ M $ are constant, equal to 1?
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2 Answers
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Hitchin showed (here, I think) that there is an exotic sphere which admits no metric of positive scalar curvature. This manifold certainly has the sphere as its topological universal cover, but its sectional curvatures can't all be positive let alone 1.
The OP is also interested in the smooth case. I think there still are counterexamples, sometimes called "fake lens spaces" - this is a useful term to google, though many of the hits focus on the topological category.
The first step is to classify spherical space forms (i.e. manifolds with constant sectional curvature 1). By various classical results in Riemannian geometry this amounts to classifying all finite groups which act freely on the sphere by isometries. This has been done, and so your question becomes "are there any other finite groups which act freely and smoothly on the sphere?" The answer is "no" in even dimensions because the only nontrivial group which can act on an even dimensional sphere is $\mathbb{Z}/2\mathbb{Z}$ (the Euler characteristic is multiplicative for coverings). The answer is also "no" in dimension 3 by the geometrization theorem.
But more interesting things can happen in higher dimensions. The results inevitably use a lot of heavy duty surgery theory, so I'll point you to a survey paper by Hambleton which goes into more detail. He credits Lee and Petrie for producing the first smooth counterexamples.
answered Jul 1, 2018 at 20:26
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6$\begingroup$ It is not polite to change the question after a good answer. I suggest that you revert to the original, accept a good answer to your question, and ask a new question. $\endgroup$ Commented Jul 1, 2018 at 20:56
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2$\begingroup$ @LeeMosher: You are right. I thought about this but decided to edit the question all the same because Paul had replied almost immediately after the question had been posted, and because the edit consisted of 1 word only. I have followed your suggestion to revert to the original question and accept his answer which is perfectly adequate for the original question. I will however not post a new question as it is too similar to this one. I will leave the question that I had intended as a comment (see below). I apologize to everyone for my mistake in the original statement. $\endgroup$ Commented Jul 1, 2018 at 22:16
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$\begingroup$ @PaulSiegel: Thanks for your reply. Do you know a counter-example when the covering map is required to be smooth (not just continuous)? This is what I had intended to ask originally, as I didn't expect any subtlety involving exotic spheres. $\endgroup$ Commented Jul 1, 2018 at 22:19
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2$\begingroup$ @rgnrmllbrg I added some remarks on the smooth case to my answer - I believe the conclusion is that there are still counter-examples, but they use a lot of surgery theory and so I can't explain them with much certainty. $\endgroup$ Commented Jul 1, 2018 at 22:56
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$\begingroup$ @PaulSiegel: Thank you very much for your thorough reply, I really appreciate it! I will look at the paper that you mentioned, but I already got what I need from what you wrote. $\endgroup$ Commented Jul 1, 2018 at 23:02
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Updated: If $n = 3$, then the answer to your question is yes. This is essentially the Thurston Elliptization Conjecture which was settled by Perelman's proof of the Geometrization Conjecture.
In my initial answer, I had hoped to use the obstruction $\alpha$ to construct a manifold which was finitely covered by $S^n$ which did not even admit positive scalar curvature metrics. As is discussed in the comments, this construction cannot work for lens spaces. In fact, it cannot happen for $n \geq 5$ at all, as was shown in Fake spherical spaceforms of constant positive scalar curvature by Kwasik & Schultz.
In Paul Seigel's answer, he mentions that the answer to your question is no. Concretely, in Free Metacyclic Group Actions on Homotopy Spheres, Petrie shows that there is a smooth free action of $\mathbb{Z}_p\rtimes\mathbb{Z}_q$ on $S^{2q-1}$ where $p$ is odd and $q$ an odd prime. Such a quotient cannot admit a constant positive sectional curvature metric. However, by the result of Kwasik & Schultz, it does admit a constant positive scalar curvature metric.
Allow me to expand a little bit on the first part on Paul Siegel's answer before heading in a slightly different direction in the search for examples.
There is a homomorphism
$$\alpha : \Omega_n^{\text{spin}} \to KO(S^n) = \begin{cases}\mathbb{Z} & n \equiv 0\bmod 4\\ \mathbb{Z}_2 & n \equiv 1, 2 \bmod 8\\ 0 & \text{otherwise}\end{cases}$$
defined by Milnor. It was shown by Milnor and Adams that when $n = 1, 2 \bmod 8$ and $n \neq 1, 2$, there is an exotic $n$-sphere $\Sigma$ such that $\alpha(\Sigma) \neq 0$. In all other dimensions, exotic spheres have value zero.
It was later shown by Hitchin that if a closed spin manifold $X$ admits a metric of positive scalar curvature, then $\alpha(X) = 0$.
So, if $X$ is an $n$-dimensional closed spin manifold which admits a metric of positive scalar curvature and $n \equiv 1, 2 \bmod 8$, $n \neq 1, 2$, then $X\#\Sigma$ does not admit a metric of positive scalar curvature (connected sum is the addition operation in $\Omega^{\text{spin}}_n$ and $\alpha$ is a homomorphism).
If $S^n \to X$ is the $k$-fold universal cover (here $S^n$ denotes the standard smooth sphere), then the universal cover of $X\#\Sigma$ is $S^n\# k\Sigma$. The group of exotic spheres is finite (except possibly in dimension four), so let $d$ be the order of $\Sigma$. If $d \mid k$, then $S^n\# k\Sigma = S^n$ and you'd have an example of the phenomena you're looking for.
Note, this doesn't help when $n \equiv 2 \bmod 8$ as $S^n$ only covers $\mathbb{RP}^n$ which is not orientable, and hence not spin. When $n \equiv 1 \bmod 8$, then $S^n$ is the universal cover of infinitely many manifolds, e.g. lens spaces. In particular, for any $k$, there is an closed orientable manifold $X$ which has $S^n$ as its $k$-sheeted cover, all that's left to know is when such an $X$ can be chosen to be spin. I don't know how to do this off the top of my head, but the answers to this question seem promising.
I believe that examples can be constructed this way, but I have never taken the time to construct one, sorry.
answered Jul 2, 2018 at 0:00
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$\begingroup$ I haven't checked all the details, but I'm not sure this approach works. First, from Browder's result on the Kervaire invariant, $bP_{n+1}$ has order $2$ whenever $n$ is $1$ mod $8$. This implies $|\theta_n| = d$ is even in these dimensions (other than $n=1$), so $k$ must be even. Now, for any of the usual lens spaces, this means the covering $S^n\rightarrow L$ factors through $S^n\rightarrow \mathbb{R}P^n$. But in dimension $1$ mod $8$, $\mathbb{R}P^n$ is not spin. This implies none of these lens spaces are spin. (Of course, $S^n$ could cover other things....) $\endgroup$ Commented Jul 17, 2018 at 18:03
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$\begingroup$ @JasonDeVito: Sorry, I only just saw your comment. Even if $|\Theta_n|$ is even, couldn't there still be elements of odd order? $\endgroup$ Commented Jan 14, 2020 at 23:03
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$\begingroup$ I see that is misinterpreted - $d$ is the order of $\Sigma \in \theta_n$, not the order of $\theta_n$. In addition, when $k$ is odd, I think the lens spaces (in any dimension) are trivially spin, because I think for any such such space $L$, $H^\ast(L;\mathbb{Z}/2\mathbb{Z})\cong H^\ast(S^n;\mathbb{Z}/2\mathbb{Z})$. (I am also reading this a year later - so take this with a grain of salt!) $\endgroup$ Commented Jan 14, 2020 at 23:15
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$\begingroup$ @JasonDeVito: Unfortunately the approach I outlined can't work for lens spaces. If $\Sigma$ is an exotic sphere in dimension $n \equiv 1 \bmod 8$ of order $d$, then $0 = \alpha(S^n) = \alpha(d\Sigma) = d\alpha(\Sigma)$, so if $\alpha(\Sigma) \neq 0$, then $d$ is even. But then as you point out, $X$ is covered by $\mathbb{RP}^n$ which is not spin, so $X$ is not spin. $\endgroup$ Commented Jan 15, 2020 at 2:06
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$\begingroup$ There are examples of fake lens spaces that require no surgery/exotic smoothness considerations. Start from any finite cyclic group $G$ of order $5$ or $\ge 7$, so $Wh(G)\neq 0$. If $L$ is any lens space $S^n/G$ with $n\ge 5$, then there is a nontrivial $h$-cobordism with one boundary equal $L$. The other boundary $L^\prime$ is not a lens space by a theorem of Atiyah-Bott, see Milnor's "Whitehead torsion", Corollary 12.13. The universal cover of the cobordism is a trivial h-cobordism, so the universal cover of $L^\prime$ is the standard sphere. $\endgroup$ Commented Jan 19, 2020 at 17:02