Rosenberg's positive scalar curvature conjecture for odd-order fundamental groups

Let $M^n$ be a connected closed manifold whose fundamental group $\pi_1(M)$ is finite of odd order. A metric of positive scalar curvature on the universal cover of $M$ need not descend to $M$. Rosenberg's conjecture. $M$ admits a metric of positive scalar curvature if and only if its universal cover does. The statement is disproved by examples in the paper: there are manifolds with finite fundamental groups of odd order that do not admit metrics of positive scalar curvature, although their universal covers do.