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?

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    $\begingroup$ Connected sum of 2 real projective planes is the Klein bottle, it indeed has a Riemannian metric of curvature 0. $\endgroup$ – YCor May 12 '18 at 16:10
  • $\begingroup$ Once it's true for a given dimension I imagine it's going to be true for all higher dimensions, by taking products with flat tori. I believe, however, that there are still only finitely many in dimension 3: the manifolds of constant curvature +1 and 0, and $S^2 \times S^1$. $\endgroup$ – Mike Miller May 12 '18 at 16:10
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    $\begingroup$ @ Mike Miller, there are infinitely many topologically distinct $3$-manifolds that are locally isometric to $S^{3}$ $\endgroup$ – Nick L May 12 '18 at 16:38
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    $\begingroup$ clearly there are already infinitely many $L(p,q)$... a stupid comment on my part. Thanks. $\endgroup$ – Mike Miller May 12 '18 at 16:51
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    $\begingroup$ The comments of MikeMiller and NickL taken together look like they settle the question. But is the answer known in any dimensions $\ge 4$ with the additional assumption of simple connectivity? $\endgroup$ – Lee Mosher May 12 '18 at 18:03

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).

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