Negatively curved manifolds with exotic smooth structures
Authors:
F. T. Farrell and L. E. Jones
Journal:
J. Amer. Math. Soc. 2 (1989), 899908
MSC:
Primary 53C20; Secondary 57R10, 57R55, 57R67
MathSciNet review:
1002632
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Abstract: Let denote a compact real hyperbolic manifold with dimension and sectional curvature , and let be an exotic sphere of dimension . Given any small number , we show that there is a finite covering space of satisfying the following properties: the connected sum is not diffeomorphic to , but it is homeomorphic to ; supports a Riemannian metric having all of its sectional curvature values in the interval . 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 LawsonYau conjecture. Note that Mostow's Rigidity Theorem [17] implies that 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 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].
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 R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 149. 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, SpringerVerlag, Berlin and New York, 1973. MR 0420609 (54:8623a)
 [3]
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 [4]
 F. T. Farrell and W.C. Hsiang, On Novikov's Conjecture for nonpositively curved manifolds. I, Ann. of Math. (2) 113 (1981), 199209. MR 604047 (83j:57018)
 [5]
 F. T. Farrell and L. E. Jones, Anosov diffeomorphisms constructed from , Topology 17 (1978), 273282. MR 508890 (81f:58030)
 [6]
 , Compact negatively curved manifolds (of ) are topologically rigid, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), 34613463. MR 997635 (90h:57023b)
 [7]
 , A topological analogue of Mostow's Rigidity Theorem, J. Amer. Math. Soc. 2 (1989), 257370. MR 973309 (90h:57023a)
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 , Examples of expanding endomorphisms on exotic tori, Invent. Math. 45 (1978), 175179. MR 0474416 (57:14056)
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 , Hyperbolic groups, Essays in Group Theory (S. M. Gersten, ed.), Math. Sci. Res. Inst. Publ. 8 (1987), 75263. MR 919829 (89e:20070)
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 G. D. Mostow, Quasiconformal mappings in space and the rigidity of hyperbolic space forms, Inst. Hautes Études Sci. Publ. 34 (1967), 53104. MR 0236383 (38:4679)
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Additional Information
F. T. Farrell
Affiliation:
L. E. Jones
Affiliation:
DOI:
http://dx.doi.org/10.1090/S08940347198910026322
PII:
S 08940347(1989)10026322
Article copyright:
© Copyright 1989 American Mathematical Society
