Negatively curved manifolds with exotic smooth structures

Authors:
F. T. Farrell and L. E. Jones

Journal:
J. Amer. Math. Soc. **2** (1989), 899-908

MSC:
Primary 53C20; Secondary 57R10, 57R55, 57R67

MathSciNet review:
1002632

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Abstract: Let denote a compact real hyperbolic manifold with dimension and sectional curvature , and let be an exotic sphere of dimension . Given any small number , we show that there is a finite covering space of satisfying the following properties: the connected sum is not diffeomorphic to , but it is homeomorphic to ; supports a Riemannian metric having all of its sectional curvature values in the interval . Thus, there are compact Riemannian manifolds of strictly negative sectional curvature which are not diffeomorphic but whose fundamental groups are isomorphic. This answers Problem 12 of the list compiled by Yau [22]; i.e., it gives counterexamples to the Lawson-Yau conjecture. Note that Mostow's Rigidity Theorem [17] implies that does not support a Riemannian metric whose sectional curvature is identically -1 . (In fact, it is not diffeomorphic to any locally symmetric space.) Thus, the manifold supports a Riemannian metric with sectional curvature arbitrarily close to -1 , but it does not support a Riemannian metric whose sectional curvature is identically -1 . More complicated examples of manifolds satisfying the properties of the previous sentence were first constructed by Gromov and Thurston [11].

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DOI:
https://doi.org/10.1090/S0894-0347-1989-1002632-2

Article copyright:
© Copyright 1989
American Mathematical Society