Negatively curved manifolds with exotic smooth structures

Farrell, F. T. ; Jones, L. E.

Journal of the American Mathematical Society, Tome 02 (1989) no. 4, pp. 899-908

Résumé

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 [22]; i.e., it gives counterexamples to the Lawson-Yau conjecture. Note that Mostow’s Rigidity Theorem [17] 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 [11].

Détail

DOI : 10.1090/S0894-0347-1989-1002632-2

Bibliographie

[1] Bishop, R. L., O’Neill, B. Manifolds of negative curvature Trans. Amer. Math. Soc. 1969 1 49

[2] Boardman, J. M., Vogt, R. M. Homotopy invariant algebraic structures on topological spaces 1973

[3] Brumfiel, G. Homotopy equivalences of almost smooth manifolds Comment. Math. Helv. 1971 381 407

[4] Farrell, F. T., Hsiang, W. C. On Novikov’s conjecture for nonpositively curved manifolds. I Ann. of Math. (2) 1981 199 209

[5] Farrell, F. T., Jones, L. E. Anosov diffeomorphisms constructed from 𝜋₁𝐷𝑖𝑓𝑓(𝑆ⁿ) Topology 1978 273 282

[6] Farrell, F. T., Jones, L. E. A topological analogue of Mostow’s rigidity theorem J. Amer. Math. Soc. 1989 257 370

[7] Farrell, F. T., Jones, L. E. A topological analogue of Mostow’s rigidity theorem J. Amer. Math. Soc. 1989 257 370

[8] Farrell, F. T., Jones, L. E. Examples of expanding endomorphisms on exotic tori Invent. Math. 1978 175 179

[9] Gromov, M. Manifolds of negative curvature J. Differential Geometry 1978 223 230

[10] Gromov, M. Hyperbolic groups 1987 75 263

[11] Gromov, M., Thurston, W. Pinching constants for hyperbolic manifolds Invent. Math. 1987 1 12

[12] Hamenstã¤Dt, Ursula A geometric characterization of negatively curved locally symmetric spaces J. Differential Geom. 1991 193 221

[13] Hicks, Noel J. Notes on differential geometry 1965

[14] Kervaire, Michel A., Milnor, John W. Groups of homotopy spheres. I Ann. of Math. (2) 1963 504 537

[15] Kirby, Robion C., Siebenmann, Laurence C. Foundational essays on topological manifolds, smoothings, and triangulations 1977

[16] Magnus, W. Residually finite groups Bull. Amer. Math. Soc. 1969 305 316

[17] Mostow, G. D. Quasi-conformal mappings in 𝑛-space and the rigidity of hyperbolic space forms Inst. Hautes Études Sci. Publ. Math. 1968 53 104

[18] Mostow, G. D., Siu, Yum Tong A compact Kähler surface of negative curvature not covered by the ball Ann. of Math. (2) 1980 321 360

[19] Ruh, Ernst A. Almost symmetric spaces 1984 93 98

[20] Siu, Yum Tong The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds Ann. of Math. (2) 1980 73 111

[21] Sullivan, Dennis Hyperbolic geometry and homeomorphisms 1979 543 555

[22] Seminar on Differential Geometry 1982

[23] Al′Ber, S. I. Spaces of mappings into a manifold of negative curvature Dokl. Akad. Nauk SSSR 1968 13 16

[24] Eells, James, Jr., Sampson, J. H. Harmonic mappings of Riemannian manifolds Amer. J. Math. 1964 109 160

[25] Hartman, Philip On homotopic harmonic maps Canadian J. Math. 1967 673 687