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Our goal here is to study compact manifolds with positive \\ $\\Gamma_2$-curvature, \\ i.e., when $\\sigma_1(g)>0$ and $\\sigma_2(g)>0$. In particular, we prove that a 3-connected non-string manifold $M$ admits a positive$\\Gamma_2$-curvature metric if and only if it admits a positive scalar curvature metric. 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