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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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 $ M$ denote a compact real hyperbolic manifold with dimension $ m \geq 5$ and sectional curvature $ K = - 1$, and let $ \Sigma $ be an exotic sphere of dimension $ m$. Given any small number $ \delta > 0$, we show that there is a finite covering space $ \widehat{M}$ of $ M$ satisfying the following properties: the connected sum $ \widehat{M}\char93 \Sigma $ is not diffeomorphic to $ \widehat{M}$, but it is homeomorphic to $ \widehat{M}$; $ \widehat{M}\char93 \Sigma $ supports a Riemannian metric having all of its sectional curvature values in the interval $ [ - 1 - \delta , - 1 + \delta ]$. 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 $ \widehat{M}\char93 \Sigma $ 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 $ \widehat{M}\char93 \Sigma $ 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: http://dx.doi.org/10.1090/S0894-0347-1989-1002632-2
PII: S 0894-0347(1989)1002632-2
Article copyright: © Copyright 1989 American Mathematical Society