It is well known that there exists three and four manifolds that do not admit an Einstein metric, but I wonder if this question is still open for manifolds of dimension higher than four. That is, does anyone know of a compact n-manifold, $n>4$, that does not admit an Einstein metric?



Obstructions to Einstein metrics in high dimensions
No one knows this. Here is a citation from Gromov's beautiful article: <http://www.ihes.fr/~gromov/topics/SpacesandQuestions.pdf>


Page 19.


*Following Alex we (I speak for myself) are lead to the pessimistic conclusion that there is no chance for a distinguished* $g\_{best}$ *for* $n\ge 5$ *and "natural" metrics, e.g. Einstein $G$, $Ri(g)=\lambda g$ for $\lambda<0$, must be chaotically scattered in the vastness of $G$ with no meaningful link between geometry and topology (This does not preclude, but rather predicts, the existence of such metrics, e.g. Einstein, on all $V$ if dimension $\ge 5$: the problem is there may be too many of them)*


