The double of a hyperbolic manifold and nonpositively curved exotic structures
Author:
Pedro Ontaneda
Journal:
Trans. Amer. Math. Soc. 355 (2003), 935965
MSC (2000):
Primary 53C20, 57Q25, 57R55
Published electronically:
October 29, 2002
MathSciNet review:
1938740
Fulltext PDF Free Access
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Abstract: We give examples of noncompact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three nonequivalent smoothable structures, two of which admit a Riemannian metric of nonpositive curvature while the third does not. We also prove that the doubles of noncompact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.
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R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitex, SpringerVerlag, New York (1991). MR 94e:57015
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R.L. Bishop and B. O'Neil, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 149. MR 40:4891
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A. Borel, Compact CliffordKlein forms of symmetric spaces, Topology 2 (1963) 111122. MR 26:3823
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A. Borel, Introduction Aux Groupes Arithmetiques, Hermann, Paris (1969). MR 39:5577
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M. Bridson and A. Haeflinger, Metric spaces of nonpositive curvature, SpringerVerlag (1999). MR 2000k:53038
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F.T. Farrell and L.E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (1989) 899908. MR 90f:53075
 8.
F.T. Farrell and L.E. Jones, Rigidity in geometry and topology, Proceedings of the International Congress of Mathematicians, Kyoto, Japan (1990). MR 93g:57041
 9.
F.T. Farrell and L.E. Jones, Exotic smoothings of hyperbolic manifolds which do not support pinched negative curvature, Proc. Amer. Math. Soc. 121 (1994) 627630. MR 94h:57047
 10.
F. T. Farrell, L. E. Jones and P. Ontaneda, Hyperbolic manifolds with negatively curved exotic triangulations in dimensions larger than five, Jour. Diff. Geom. 48 (1998) 319322. MR 2000f:57003
 11.
D. Gromoll and J.A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971) 545552. MR 43:6841
 12.
R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Math. Studies, 88, Princeton University Press, Princeton (1977). MR 58:31082
 13.
H.B. Lawson and S.T. Yau, Compact manifolds of nonpositive curvature, J. Diff. Geom. 7 (1972) 211228. MR 48:12402
 14.
J.J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. 104 (1976) 235247. MR 54:10488
 15.
P. Ontaneda, Hyperbolic manifolds with negatively curved exotic triangulations in dimension six, Jour. Diff. Geom. 40 (1994) 722. MR 95d:57013
 16.
C. P. Rourke and B. J. Sanderson, Introduction to piecewiselinear topology, SpringerVerlag, (1972). MR 50:3236
 17.
C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, London (1971). MR 55:4217
 1.
 C.S. Aravinda and F.T. Farrell, Rank 1 aspherical manifolds which do not support any nonpositively curved metric, Comm. in Analysis and Geometry 2 (1994) 6578. MR 96e:53045
 2.
 R. Benedetti and C. Petronio, Lectures on Hyperbolic Geometry, Universitex, SpringerVerlag, New York (1991). MR 94e:57015
 3.
 R.L. Bishop and B. O'Neil, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969) 149. MR 40:4891
 4.
 A. Borel, Compact CliffordKlein forms of symmetric spaces, Topology 2 (1963) 111122. MR 26:3823
 5.
 A. Borel, Introduction Aux Groupes Arithmetiques, Hermann, Paris (1969). MR 39:5577
 6.
 M. Bridson and A. Haeflinger, Metric spaces of nonpositive curvature, SpringerVerlag (1999). MR 2000k:53038
 7.
 F.T. Farrell and L.E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (1989) 899908. MR 90f:53075
 8.
 F.T. Farrell and L.E. Jones, Rigidity in geometry and topology, Proceedings of the International Congress of Mathematicians, Kyoto, Japan (1990). MR 93g:57041
 9.
 F.T. Farrell and L.E. Jones, Exotic smoothings of hyperbolic manifolds which do not support pinched negative curvature, Proc. Amer. Math. Soc. 121 (1994) 627630. MR 94h:57047
 10.
 F. T. Farrell, L. E. Jones and P. Ontaneda, Hyperbolic manifolds with negatively curved exotic triangulations in dimensions larger than five, Jour. Diff. Geom. 48 (1998) 319322. MR 2000f:57003
 11.
 D. Gromoll and J.A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971) 545552. MR 43:6841
 12.
 R.C. Kirby and L.C. Siebenmann, Foundational Essays on Topological Manifolds, Smoothings, and Triangulations, Annals of Math. Studies, 88, Princeton University Press, Princeton (1977). MR 58:31082
 13.
 H.B. Lawson and S.T. Yau, Compact manifolds of nonpositive curvature, J. Diff. Geom. 7 (1972) 211228. MR 48:12402
 14.
 J.J. Millson, On the first Betti number of a constant negatively curved manifold, Ann. of Math. 104 (1976) 235247. MR 54:10488
 15.
 P. Ontaneda, Hyperbolic manifolds with negatively curved exotic triangulations in dimension six, Jour. Diff. Geom. 40 (1994) 722. MR 95d:57013
 16.
 C. P. Rourke and B. J. Sanderson, Introduction to piecewiselinear topology, SpringerVerlag, (1972). MR 50:3236
 17.
 C.T.C. Wall, Surgery on Compact Manifolds, Academic Press, London (1971). MR 55:4217
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Additional Information
Pedro Ontaneda
Affiliation:
Departamento de Matematica, Universidade Federal de Pernambuco, Cidade Universitaria, Recife, PE 50670901, Brazil
Email:
ontaneda@dmat.ufpe.br
DOI:
http://dx.doi.org/10.1090/S0002994702030763
PII:
S 00029947(02)030763
Received by editor(s):
April 12, 2001
Received by editor(s) in revised form:
April 12, 2002
Published electronically:
October 29, 2002
Additional Notes:
This research was supported in part by CNPq, Brazil
Article copyright:
© Copyright 2002
American Mathematical Society
