  Gromov showed that there is an upper bound on the Betti numbers of all closed
Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether
such manifolds (if simply connected) fall into only finitely many rational
homotopy types. We give a negative answer, in fact in dimension 6, which is the
smallest possible. We also give counterexamples to some related questions in
dimensions 7 and 9, improving the original counterexamples by Fang and Rong
which were in dimensions at least 22.
