# Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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## Negatively curved manifolds with exotic smooth structuresHTML articles powered by AMS MathViewer

by F. T. Farrell and L. E. Jones
J. Amer. Math. Soc. 2 (1989), 899-908 Request permission

## 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 $\widehat {M}\# \Sigma$ is not diffeomorphic to $\widehat {M}$, but it is homeomorphic to $\widehat {M}$; $\widehat {M}\# \Sigma$ 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 ; i.e., it gives counterexamples to the Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem  implies that $\widehat {M}\# \Sigma$ 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 $\widehat {M}\# \Sigma$ 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 .
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