Negatively curved manifolds with exotic smooth structures

Abstract

Let M M denote a compact real hyperbolic manifold with dimension m ≥ 5 m \geq 5 and sectional curvature K = − 1 K = - 1 , and let Σ \Sigma be an exotic sphere of dimension m m . Given any small number δ > 0 \delta > 0 , we show that there is a finite covering space M ^ \widehat {M} of M M satisfying the following properties: the connected sum M ^ # Σ \widehat {M}\# \Sigma is not diffeomorphic to M ^ \widehat {M} , but it is homeomorphic to M ^ \widehat {M} ; M ^ # Σ \widehat {M}\# \Sigma supports a Riemannian metric having all of its sectional curvature values in the interval [ − 1 − δ , − 1 + δ ] [ - 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 M ^ # Σ \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 M ^ # Σ \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 [11].