An essential relation between Einstein metrics, volume entropy, and exotic smooth structures (Q843066)

Following a terminology introduced by \textit{M. Gromov} in [J. Differ. Geom. 18, 1--147 (1983; Zbl 0515.53037)], it is said that a closed connected manifold \(V\) is essential if for some map \(f:V\rightarrow K\) into an aspherical space the induced top-dimensional homomorphism on homology does not vanish. The first main result of the paper shows that the minimal volume entropy of a connected closed manifold \(M\) doesn't change for the connected sum of \(M\) with a connected closed orientable manifold \(N\) which is not essential. This result is used to extract new information about exotic smooth structures on 4-manifolds and existence of Einstein metrics. The following main result of the paper gives a positive answer to an open question which stems from work of \textit{C. LeBrun} in [Math. Res. Lett. 3, No.~2, 133--147 (1996; Zbl 0856.53035)]. The authors show that there exists an infinite family of 4-manifolds with the following properties: {\parindent=5mm \begin{itemize}\item[1)] They have positive minimal volume entropy. \item[2)] They satisfy a strict version of the Gromov-Hitchin-Thorpe inequality with a minimal volume entropy term. This implies that their homotopy type satisfies all the restrictions known so far to the existence of an Einstein metric. \item[3)] They nevertheless each admit infinitely many distinct smooth structures for which no compatible Einstein metric exists. \end{itemize}} This result is obtained by using Seiberg-Witten theory and the stable cohomotopy invariant of \textit{S. Bauer} and \textit{M. Furuta} [Invent. Math. 155, No. 1, 1--19 (2004; Zbl 1050.57024)].