## Negatively curved manifolds with exotic smooth structures

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- by F. T. Farrell and L. E. Jones
- J. Amer. Math. Soc.
**2**(1989), 899-908 - DOI: https://doi.org/10.1090/S0894-0347-1989-1002632-2
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## 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 [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].## References

- R. L. Bishop and B. O’Neill,
*Manifolds of negative curvature*, Trans. Amer. Math. Soc.**145**(1969), 1–49. MR**251664**, DOI 10.1090/S0002-9947-1969-0251664-4 - J. M. Boardman and R. M. Vogt,
*Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics, Vol. 347, Springer-Verlag, Berlin-New York, 1973. MR**0420609** - G. Brumfiel,
*Homotopy equivalences of almost smooth manifolds*, Comment. Math. Helv.**46**(1971), 381–407. MR**305419**, DOI 10.1007/BF02566851 - F. T. Farrell and W. C. Hsiang,
*On Novikov’s conjecture for nonpositively curved manifolds. I*, Ann. of Math. (2)**113**(1981), no. 1, 199–209. MR**604047**, DOI 10.2307/1971138 - F. T. Farrell and L. E. Jones,
*Anosov diffeomorphisms constructed from $\pi _{1}\,\textrm {Diff}\,(S^{n})$*, Topology**17**(1978), no. 3, 273–282. MR**508890**, DOI 10.1016/0040-9383(78)90031-9 - F. T. Farrell and L. E. Jones,
*A topological analogue of Mostow’s rigidity theorem*, J. Amer. Math. Soc.**2**(1989), no. 2, 257–370. MR**973309**, DOI 10.1090/S0894-0347-1989-0973309-4 - F. T. Farrell and L. E. Jones,
*A topological analogue of Mostow’s rigidity theorem*, J. Amer. Math. Soc.**2**(1989), no. 2, 257–370. MR**973309**, DOI 10.1090/S0894-0347-1989-0973309-4 - F. T. Farrell and L. E. Jones,
*Examples of expanding endomorphisms on exotic tori*, Invent. Math.**45**(1978), no. 2, 175–179. MR**474416**, DOI 10.1007/BF01390271 - M. Gromov,
*Manifolds of negative curvature*, J. Differential Geometry**13**(1978), no. 2, 223–230. MR**540941** - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - M. Gromov and W. Thurston,
*Pinching constants for hyperbolic manifolds*, Invent. Math.**89**(1987), no. 1, 1–12. MR**892185**, DOI 10.1007/BF01404671 - Ursula Hamenstädt,
*A geometric characterization of negatively curved locally symmetric spaces*, J. Differential Geom.**34**(1991), no. 1, 193–221. MR**1114460** - Noel J. Hicks,
*Notes on differential geometry*, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR**0179691** - Michel A. Kervaire and John W. Milnor,
*Groups of homotopy spheres. I*, Ann. of Math. (2)**77**(1963), 504–537. MR**148075**, DOI 10.1090/S0273-0979-2015-01504-1 - Robion C. Kirby and Laurence C. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*, Annals of Mathematics Studies, No. 88, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1977. With notes by John Milnor and Michael Atiyah. MR**0645390** - W. Magnus,
*Residually finite groups*, Bull. Amer. Math. Soc.**75**(1969), 305–316. MR**241525**, DOI 10.1090/S0002-9904-1969-12149-X - G. D. Mostow,
*Quasi-conformal mappings in $n$-space and the rigidity of hyperbolic space forms*, Inst. Hautes Études Sci. Publ. Math.**34**(1968), 53–104. MR**236383** - G. D. Mostow and Yum Tong Siu,
*A compact Kähler surface of negative curvature not covered by the ball*, Ann. of Math. (2)**112**(1980), no. 2, 321–360. MR**592294**, DOI 10.2307/1971149 - Ernst A. Ruh,
*Almost symmetric spaces*, Global Riemannian geometry (Durham, 1983) Ellis Horwood Ser. Math. Appl., Horwood, Chichester, 1984, pp. 93–98. MR**757210** - Yum Tong Siu,
*The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds*, Ann. of Math. (2)**112**(1980), no. 1, 73–111. MR**584075**, DOI 10.2307/1971321 - Dennis Sullivan,
*Hyperbolic geometry and homeomorphisms*, Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977) Academic Press, New York-London, 1979, pp. 543–555. MR**537749** - Shing Tung Yau (ed.),
*Seminar on Differential Geometry*, Annals of Mathematics Studies, No. 102, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars held during the academic year 1979–1980. MR**645728** - S. I. Al′ber,
*Spaces of mappings into a manifold of negative curvature*, Dokl. Akad. Nauk SSSR**178**(1968), 13–16 (Russian). MR**0230254** - James Eells Jr. and J. H. Sampson,
*Harmonic mappings of Riemannian manifolds*, Amer. J. Math.**86**(1964), 109–160. MR**164306**, DOI 10.2307/2373037 - Philip Hartman,
*On homotopic harmonic maps*, Canadian J. Math.**19**(1967), 673–687. MR**214004**, DOI 10.4153/CJM-1967-062-6

## Bibliographic Information

- © Copyright 1989 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**2**(1989), 899-908 - MSC: Primary 53C20; Secondary 57R10, 57R55, 57R67
- DOI: https://doi.org/10.1090/S0894-0347-1989-1002632-2
- MathSciNet review: 1002632