Compact negatively curved manifolds (of dim [unk] 3,4) are topologically rigid.
Let M be a complete (connected) Riemannian manifold having finite volume and whose sectional curvatures lie in the interval [c(1), c(2)] with -infinity < c(1)[unk]c(2) < 0. Then any proper homotopy equivalence h:N --> M from a topological manifold N is properly homotopic to a homeomorphism, provided the dimension of M is >5. In particular, if M and N are both compact (connected) negatively curved Riemannian manifolds with isomorphic fundamental groups, then M and N are homeomorphic provided dim M [unk] 3 and 4. {If both are locally symmetric, this is a consequence of Mostow's rigidity theorem [Mostow, G. D. (1967) Publ. Inst. Haut. Etud. Sci. 34, 53-104].} When M has infinite volume we can still calculate the surgery L-groups of pi(1)M, even when dim M = 3, 4, or 5, provided M is locally symmetric. An identification of the weak homotopy type of the homeomorphism group of (finite volume) M is also made through a stable range. We have previously announced these results for the special case that c(1) = c(2) = -1.
Department of Mathematics, Columbia University, New York, NY 10027.
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Jones