A (special case of a) theorem of Gromov says for any $n\in \mathbb{N}$ there exists a constant $C(n)$ such that for any smooth connected closed $n$-dimensional Riemannian manifold with non-negative sectional curvature the sum of all of its Betti numbers is at most $C(n)$.

On the other hand, for $n=2$ only sphere, torus, real projective plane, and (possibly- I do not know) connected sum of 2 real projective planes, can carry a non-negatively curved metric. (This follows from the Gauss-Bonnet theorem and topological classification of surfaces with non-negative Euler characteristic.)

What happens when $n>2$? For which $n>2$ it is known that there are infinitely many homeomorphism/ homotopy types of $n$-manifolds which can carry a Riemannian metric with non-negative sectional curvature?