Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).