The double of a hyperbolic manifold and non-positively curved exotic structures

Author:
Pedro Ontaneda

Journal:
Trans. Amer. Math. Soc. **355** (2003), 935-965

MSC (2000):
Primary 53C20, 57Q25, 57R55

DOI:
https://doi.org/10.1090/S0002-9947-02-03076-3

Published electronically:
October 29, 2002

MathSciNet review:
1938740

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Abstract | References | Similar Articles | Additional Information

Abstract: We give examples of non-compact finite volume real hyperbolic manifolds of dimension greater than five, such that their doubles admit at least three non-equivalent smoothable structures, two of which admit a Riemannian metric of non-positive curvature while the third does not. We also prove that the doubles of non-compact finite volume real hyperbolic manifolds of dimension greater than four are differentiably rigid.

**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) 65-78. MR**96e:53045****2.**R. Benedetti and C. Petronio,*Lectures on Hyperbolic Geometry*, Universitex, Springer-Verlag, New York (1991). MR**94e:57015****3.**R.L. Bishop and B. O'Neil,*Manifolds of negative curvature*, Trans. Amer. Math. Soc.**145**(1969) 1-49. MR**40:4891****4.**A. Borel,*Compact Clifford-Klein forms of symmetric spaces*, Topology**2**(1963) 111-122. 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 non-positive curvature*, Springer-Verlag (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) 899-908. 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) 627-630. 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) 319-322. 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 non-positive curvature*, Bull. Amer. Math. Soc.**77**(1971) 545-552. 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) 211-228. MR**48:12402****14.**J.J. Millson,*On the first Betti number of a constant negatively curved manifold*, Ann. of Math.**104**(1976) 235-247. MR**54:10488****15.**P. Ontaneda,*Hyperbolic manifolds with negatively curved exotic triangulations in dimension six*, Jour. Diff. Geom.**40**(1994) 7-22. MR**95d:57013****16.**C. P. Rourke and B. J. Sanderson,*Introduction to piecewise-linear topology*, Springer-Verlag, (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 50670-901, Brazil

Email:
ontaneda@dmat.ufpe.br

DOI:
https://doi.org/10.1090/S0002-9947-02-03076-3

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