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We prove that the second Betti number of a compact Riemannian manifold vanishes under certain Ricci curved restriction. As consequences we obtain an interesting curved restriction for compact Kähler-Einstein manifolds and a homology sphere theorem in dim = 4, 5.doi:10.4310/ajm.2014.v18.n4.a2 fatcat:phigqya6irhodhiehw54ms7sfq