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].

The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory

“…The Borel Conjecture 1.27 is false in the smooth category, i.e. if one replaces topological manifold by smooth manifold and homeomorphism by diffeomorphism [106].…”

Symmetries of exotic negatively curved manifolds

“…In any case, if M does not admit an orientation-reversing isometry, then Ψ N is surjective. Farrell-Jones [FJ89a] show (implicitly) that reversing orientation is an obstruction to belonging to Im Ψ N when 2Σ = 0. According to Theorem 1, this is the only obstruction.…”

“…To prove Theorem C, one would like to promote the action of Out + (π) on N = M #Σ produced in Theorem 2 to an action by isometries with respect to some negatively curved metric on N . Using a warped-metric construction of Farrell-Jones [FJ89a], it suffices to find an M that is stably parallelizable, has large injectivity radius, and such that Isom + (M ) acts freely on M . Arranging all of these conditions simultaneously becomes delicate, especially arranging that M is stably parallelizable (which is desired because it guarantees that M #Σ is not diffeomorphic to M ).…”

Untitled

“…In other words, if M is a compact Riemannian manifold whose sectional curvatures are sufficiently close to −1, can we conclude that M is isometric to a quotient of hyperbolic space by standard isometries? This question was resolved in 1987 by M. Gromov and W. Thurston [64] (see also [52], [101]):…”