An obstruction to the existence of Einstein metrics on 4-manifolds (Q1264325)

Let \(Y\) be 4-dimensional, compact and oriented Riemannian manifold. There exist two well known topological obstructions to the existence of Einstein metrics on \(Y\). These obstructions are expressed with the help of the Euler characteristic \(\chi(Y)\), the signature \(\tau(Y)\) and the simplicial volume \(\| Y\| \) of \(Y\). The first one (Thorpe's inequality) states that if \(\chi(Y) < \frac{3}{2}| \tau(Y)| \) then \(Y\) does not admit any Einstein metric. The second one (Gromov's inequality) shows that if \(\chi(Y)< \frac{1}{2592{\pi}^2}\| Y\| \) then \(Y\) does not admit any Einstein metric. In the paper under review, the author shows that, under the topological hypothesis that \(Y\) admits a nonzero degree map on some compact locally symmetric 4-manifold, additional obstructions arise on \(\chi(Y)\) and \(\tau(Y)\). Some of these obstructions are sharp inequalities, that is, the equality case contains one and only one Einstein manifold (the locally symmetric one). The author's new obstructions produce a lot of 4-manifolds which admit no Einstein metric and, nevertheless, do not satisfy Thorpe's and Gromov's inequalities.