Is Gromov's conjecture about optimal Betti number estimate true for compact manifolds? Gromov's conjecture state that:

Conjecture: Any complete manifold $(M, g)$ with $\sec\geq 0$ and for any field $\Bbb F$ of coefficients satisfies
$$\sum_{i=0}^n b_i(M,\Bbb F)\leq 2^n.$$

Is Gromov's conjecture true for compact non-negatively curved manifolds?
 A: Gromov's conjecture is unknown for closed manifolds of non-negative sectional curvature.  It is even unknown for positively curved closed manifolds specifically in the case where the field is the rational numbers!
At least for rational coefficients, Gromov's conjecture for closed manifolds would follow from the Bott conjecture (sometimes called the Bott-Grove-Halperin conjecture) which asserts that a closed Riemannianian manifold of non-negative sectional curvature should be rationally elliptic.
Lastly, let me just point out that the non-compact case case is a consequence of the closed case in the following sense:

Suppose Gromov's conjecture is true for every closed non-negatively curved manifold of dimension at most $n$.  If $M$ is a non-compact complete non-negatively curved Riemannian manifold of dimension $n+1$, then $\sum_{i=0}^{n+1} b_i(M;\mathbb{F})\leq 2^n$.

In other words, from the perspective of Betti numbers, a non-compact manifold behaves like a closed manifold of smaller dimension.  This result is an easy consequence of Cheeger and Gromoll's Soul Theorem, which asserts that if $M$ is a non-negatively curved complete non-compact Riemannian manifold, then it has a closed totally geodesic submanifold $S$ (called the soul) for which $M$ is diffeomorphic to the normal bundle of $S$.  In particular, $M$ deformation retracts to a smaller dimensional closed submanifold.
