Negatively curved manifolds with exotic smooth structures

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].