Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Negatively curved manifolds with exotic smooth structures


Authors: F. T. Farrell and L. E. Jones
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}\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].


References [Enhancements On Off] (What's this?)

  • [1] R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. MR 0251664 (40:4891)
  • [2] J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Math., no. 347, Springer-Verlag, Berlin and New York, 1973. MR 0420609 (54:8623a)
  • [3] G. Brumfiel, Homotopy equivalence of almost smooth manifolds, Comment. Math. Helv. 46 (1971), 381-407. MR 0305419 (46:4549)
  • [4] F. T. Farrell and W.-C. Hsiang, On Novikov's Conjecture for non-positively curved manifolds. I, Ann. of Math. (2) 113 (1981), 199-209. MR 604047 (83j:57018)
  • [5] F. T. Farrell and L. E. Jones, Anosov diffeomorphisms constructed from $ {\pi _1}\;{\text{Diff}}({S^n})$, Topology 17 (1978), 273-282. MR 508890 (81f:58030)
  • [6] -, Compact negatively curved manifolds (of $ {\text{dim}} \ne 3,4$) are topologically rigid, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 3461-3463. MR 997635 (90h:57023b)
  • [7] -, A topological analogue of Mostow's Rigidity Theorem, J. Amer. Math. Soc. 2 (1989), 257-370. MR 973309 (90h:57023a)
  • [8] -, Examples of expanding endomorphisms on exotic tori, Invent. Math. 45 (1978), 175-179. MR 0474416 (57:14056)
  • [9] M. Gromov, Manifolds of negative curvature, J. Differential Geom. 13 (1978), 223-230. MR 540941 (80h:53040)
  • [10] -, Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Math. Sci. Res. Inst. Publ. 8 (1987), 75-263. MR 919829 (89e:20070)
  • [11] M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), 1-12. MR 892185 (88e:53058)
  • [12] U. Hamenstadt, A geometric characterization of negatively curved locally symmetric spaces (to appear). MR 1114460 (92i:53046)
  • [13] N. Hicks, Notes on differential geometry, van Nostrand, Princeton, NJ, 1965. MR 0179691 (31:3936)
  • [14] M. Kervaire and J. Milnor, Groups of homotopy spheres: I, Ann. of Math. (2) 77 (1963), 504-537. MR 0148075 (26:5584)
  • [15] R. C. Kirby and L. C. Siebenmann, Foundational essays on topological manifolds, smoothings, and triangulations, Ann of Math. Stud., no. 88, Princeton Univ. Press, Princeton, NJ, 1977. MR 0645390 (58:31082)
  • [16] W. Magnus, Residually finite groups, Bull. Amer. Math. Soc. 75 (1969), 305-316. MR 0241525 (39:2865)
  • [17] G. D. Mostow, Quasi-conformal mappings in $ n$-space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. 34 (1967), 53-104. MR 0236383 (38:4679)
  • [18] G. D. Mostow and Y. T. Siu, A compact Kähler surface of negative curvature not covered by the ball, Ann. of Math. 112 (1980), 321-360. MR 592294 (82f:53075)
  • [19] E. A. Ruh, Almost symmetric spaces, Global Riemannian Geometry, Horwood, Chichester, England, 1984, pp. 93-98. MR 757210
  • [20] Y. T. Siu, The complex-analyticity of harmonic maps and the strong rigidity of compact Kähler manifolds, Ann. of Math. (2) 112 (1980), 73-111. MR 584075 (81j:53061)
  • [21] D. Sullivan, Hyperbolic geometry and homeomorphisms, Geometric Topology (J. Cantrell, ed.), Academic Press, New York, 1979, pp. 543-555. MR 537749 (81m:57012)
  • [22] S.-T. Yau, Seminar on differential geometry, Ann. of Math. Stud., no. 102, Princeton Univ. Press, Princeton, NJ, 1982. MR 645728 (83a:53002)
  • [23] S. I. Al'ber, Spaces of mappings into manifold of negative curvature, Dokl. Akad. Nauk. SSSR 178 (1968), 13-16. MR 0230254 (37:5817)
  • [24] J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 0164306 (29:1603)
  • [25] P. Hartman, On homotopic harmonic maps, Canad. J. Math. 19 (1967), 673-687. MR 0214004 (35:4856)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 53C20, 57R10, 57R55, 57R67

Retrieve articles in all journals with MSC: 53C20, 57R10, 57R55, 57R67


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1989-1002632-2
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society