If $M$ is a Ricci-flat closed Riemannian manifold with $H^1(M,\mathbb R^n)=0$, can we show the fundamental group of $M$ is finite?
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1No, we cannot, because it would be false even for flat manifolds. Google "Hantzsche-Wendt manidolds." – Moishe Kohan Oct 10 '14 at 21:41