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

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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 Lawson-Yau 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].

**[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*, Topology**17**(1978), 273-282. MR**508890 (81f:58030)****[6]**-,*Compact negatively curved manifolds (of**) 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**-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)**

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DOI:
https://doi.org/10.1090/S0894-0347-1989-1002632-2

Article copyright:
© Copyright 1989
American Mathematical Society