Exotic smooth structures on negatively curved manifolds that are not of the homotopy type of a locally symmetric space
by
Sergio Ardanza-Trevijano
SUNY Binghamton, USA

Poster

Gromov and Thurston constructed in Pinching constants for hyperbolic Manifolds, Invent. Math. 89, 1-12 Springer-Verlag (1987), examples of s-sheeted branched covers \rho:M_s --> M where M is an n-dimensional hyperbolic manifold (n > 3), M_s admits a metric of negative sectional curvature and does not have the homotopy type of a locally symmetric space.

In this poster, we introduce the result that when n=4k-1,  k > 1 there are exotic n-spheres \Sigma such that M_s # \Sigma is homeomorphic but not diffeomorphic to M_s and M_s # \Sigma admits a metric of negative curvature.

These examples of exotic smooth structures on negatively curved manifolds are different from the ones obtained by Farrell and Jones in Negative curved manifolds with exotic smooth structures, J. Amer. Math. Soc. Vol. 2 Number 4, 889-908, Providence, R.I. (1989), which had the homotopy type of a locally symmetric space. We used their methods to prove that M_s # \Sigma admits a metric of negative curvature and results by W. Browder to see that M_s # \Sigma is not diffeomorphic to M_s extracted from On the action of \Theta(\partial\pi), Differential and Combinatorial Topology (A symposium in honor of Marston Morse) pp 23-26 Princeton Univ. Press (1965).

Date received: September 12, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cafh-46.