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Let $M$ be a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$. Is $M$ diffeomorphic to a spherical space form?

I know, by a theorem of Brendle and Schoen, if $M$ is a compact Riemannian manifold of dimension $n>3$ with pointwise $1/4$-pinched sectional curvature, then $M$ is diffeomorphic to a spherical space form. Therefore, I may ask this question: does a compact Riemannian manifold with positive sectional curvature whose universal covering space is diffeomorphic to $S^n$ admit a metric with pointwise $1/4$-pinched sectional curvature?

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    $\begingroup$ I am quite sure that the answer is unknown. $\endgroup$ Commented Nov 12, 2013 at 7:30

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As Anton writes, this is unknown.

The main issue is that as of now, the only method we have of creating positively curved closed manifolds with non-trivial fundamental group is to start with a simply connected example and quotient by a free isometric action.

The only positively curved metrics on spheres which we understand well enough to carry this out are induced by invariant metrics on Lie groups. In particular, the only quotients which inherit positive curvature from this process are space forms.

If you weaken your question by changing "positive" to "non-negative", then we know a bit more. For example, Theorem G of

Grove, Karsten, and Wolfgang Ziller. “Curvature and Symmetry of Milnor Spheres.” Annals of Mathematics 152, no. 1 (2000): 331–67. arXiv version

asserts that all four oriented diffeomorphism types of $\mathbb{R}P^5$s admit metrics of non-negative sectional curvature. Since there are no exotic $5$-spheres, these examples meet your criteria (except for merely having non-negative sectional curvature).

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    $\begingroup$ dro.dur.ac.uk/37462/1/37462.pdf by Goette, Kerin, and Shankar contains a construction of non-negative sectional curvature of many cyclic quotients of homotopy $7$-spheres. If one restricts to when the homotopy sphere is the standard sphere, one obtains infinitely many examples which are not obviously diffeo to space forms, but I don't know how to prove it. Galaz-García, Kerin, and myself arxiv.org/abs/2008.10278 have shown that the Grove-Ziller and the Goette-Kerin-Shankar constructions can only possibly work in dimension $2,3,4,5,7$. $\endgroup$ Commented Dec 27, 2022 at 23:24

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