Rational Homology 5-Spheres with Positive Ricci Curvature

We prove that for every integer k>1 there is a simply connected rational homology 5-sphere $M_k^5$ with spin such that $H_2(M_k^5,\text{bbz})$ has order $k^2,$ and $M_k^5$ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of $k$ has the form $k=p_1\mathrm{.}\mathrm{.}\mathrm{.}{p}_r$ for distinct primes $p_i$ then $M_k^5$ is uniquely determined.