Curvature, diameter, and quotient manifolds
  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. 
Burt Totaro