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An essential relation between Einstein metrics, volume entropy, and exotic smooth structures. (English) Zbl 1183.53038
Following a terminology introduced by 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 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:
1)
They have positive minimal volume entropy.
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.
3)
They nevertheless each admit infinitely many distinct smooth structures for which no compatible Einstein metric exists.
This result is obtained by using Seiberg-Witten theory and the stable cohomotopy invariant of S. Bauer and M. Furuta [Invent. Math. 155, No. 1, 1–19 (2004; Zbl 1050.57024)].

MSC:
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
57R55 Differentiable structures in differential topology
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
57R57 Applications of global analysis to structures on manifolds
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