Price inequalities and Betti number growth on manifolds without conjugate points

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well of non-positive sectional curvature, this case we prove strengthened inequality. employ these to study asymptotic behavior Betti numbers coverings Riemannian points. Finally, give vanishing result $L^{2}$-Betti closed