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This is a follow up question inspired by

Fundamental groups of compact manifolds with non-negative Ricci curvature.

In dimensions 3 and 2 (and 1) a manifold has a virtually abelian fundamental group if and only if it admits a metric with nonnegative curvature. Is that true in every dimension?

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  • $\begingroup$ Here you mean sectional curvature? $\endgroup$
    – YCor
    Commented Dec 3, 2021 at 13:16
  • $\begingroup$ Yes although in general I welcome information about both sectional and Ricci curvature $\endgroup$
    – Ian Gershon Teixeira
    Commented Dec 3, 2021 at 15:36
  • $\begingroup$ In dimension 4, what about the connected sum of two copies of $S^2\times S^2$? This is simply connected. $\endgroup$
    – YCor
    Commented Dec 3, 2021 at 15:44
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    $\begingroup$ Ya I was also thinking about $ K_3 $ surface. It is simply connected and I have no idea what kind of metric you can put on that. I guess the answer to my question is probably no... Do you think there is any way to rephrase the question to get something interesting about the converse of the theorem in the linked question? $\endgroup$
    – Ian Gershon Teixeira
    Commented Dec 3, 2021 at 15:52
  • $\begingroup$ $K_3$ carries Ricci flat metrics, and we know that the scalar curvature cannot be positive by Lichnerowicz's argument. Because it cannot carry a flat metric, nonnegative sectional curvature is ruled out. $\endgroup$
    – Sebastian Goette
    Commented Dec 3, 2021 at 18:16

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