TITLE: Homeomorphism/ homotopy types of non-negatively curved manifolds
QUESTION [8 upvotes]: 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?

REPLY [14 votes]: As mentioned in comments, the first dimension where an infinite family of pairwise non-homeomorphic closed nonnegatively curved manifolds occurs is $3$ (the lens spaces). The question becomes more challenging for simply-connected manifolds. If (as expected) simply-connected nonnegatively curved manifolds are rationally elliptic, then the first infinite family occurs in dimension 6, see 
Curvature, diameter, and quotient manifolds by
B. Totaro.  In general, the question is well-studied and there are many examples (easily found by searching online).