Rational Homology 5-Spheres with Positive Ricci Curvature

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We prove that for every integer k>1 there is a simply connected rational homology 5-sphere M⁵ₖ with spin such that H₂(M⁵ₖ,) has order k², and M⁵ₖ admits a Riemannian metric of positive Ricci curvature. Moreover, if the prime number decomposition of k has the form k=p₁... pᵣ for distinct primes pᵢ then M⁵ₖ is uniquely determined.