Topological Rigidity for Hyperbolic Manifolds

Let M be a complete Riemannian manifold having constant sectional curvature —1 and finite volume. Let M denote its Gromov-Margulis manifold compactification and assume that the dimension of M is greater than 5. (If M is compact, then M = M and dM is empty.) We announce (among other results) that any homotopy equivalence h: (N,dN) —• (M,dM), where N is a compact manifold, is homotopic to a homeomorphism. This is a topological analogue of Mostow's rigidity theorem [18]. Moreover, for each integer j , the surgery group LJ(TTIM) is isomorphic to the set of homotopy classes of maps [I x M rel d, G/Top] where k is any positive integer such that k + dim M = j mod 4. Here I denotes the fc-fold product IxIx--xl where I is the closed interval [0,1]. Let M denote a complete Riemannian manifold having constant sectional curvature —1 and finite volume. Thus M is a real hyperbolic manifold of finite volume. Gromov [13] and Margulis have constructed a smooth manifold compactification of M which is denoted by M. Let I denote the fc-fold product 7 x I x x I, where i* is the closed interval [0,1]; in particular, 1° is a single point. Let TV be a compact manifold such that its boundary dN decomposes as dN — d\N U 82N where d\ AT, 82N are compact codimension zero submanifolds of dN with d{d1N) = d(d2N) = dxN C\ d2N. Set A7V = d{diN). THEOREM 1. Let h : (AT, dx AT, d2N, AN) -> {I k x M, dl x M, I x dM, dl xdM) be a homotopy equivalence ofA-tuples such that h: d\N —* (dl) x M is a homeomorphism. If k + dim(M) > 5, then there is a homotopy ht : (TV, dxN, d2N, AN) -> (I k xM, dl xM,Ix dM, dl x dM), t G [0,1], with ho = h, hi a homeomorphism and the restriction of ht to d\N the constant homotopy. Moreover, if the restriction of h to d2N is also a homeomorphism, then we need only assume that k + dim(M) > 4 and ht can be constructed so that it is constant on all of dN. COROLLARY 1. Let h: (N,dN) -» (M,dM) be a homotopy equivalence of pairs where N is a compact manifold. If dim(M) > 5, then there is a homotopy ht : (N, dN) (M, dM), t e [0,1], Received by the editors December 13, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 18F25, 22E40, 57D50. Both authors were supported in part by the NSF. ©1988 American Mathematical Society 0273-0979/88 $1.00 + $.25 per page 277 278 F. T. FARRELL AND L. E. JONES with ho = h and such that hi is a homeomorphism. Moreover, if h: dN —• dM is already a homeomorphism, then we need only assume that dim(M) > 4 and ht can be constructed so that it is constant on dN. Recall that each component of dM is homeomorphic to a flat Riemannian manifold. Hence [7] can be used to reduce the proof of Theorem 1 to the case where h: Ô2N —• I x dM is a homeomorphism. Let w\\ 7Ti(M) —• Z2 denote the homomorphism determined by the first Stiefel-Whitney class of M, and LJ(TTIM) denote the ./-dimensional surgery group for the fundamental group IÏ\M with orientation data w\, defined by Wall in [21]. Let O: [I x M reld,G/Top] -* Lfc+m(7nM) be the surgery homomorphism where m = dim M (cf. [21 and 17]). THEOREM 2. For each positive integer k, the surgery homomorphism S:[IxM reld,G/Top] -> Lk+m{inM) is an isomorphism provided fc + m > 5 . It was shown in [5 and 6] that 6 is a split monomorphism. Since LJ{TC\M) is isomorphic to Lj+±{iriM) for all integers j , Theorem 2 yields a calculation of LJ{TT\M) regardless of the dimension of M; in particular, even when M is either threeor four-dimensional. Since G/Top has the same rational homotopy type as does the product Ilj>i ^ ( Z ; 4j) of Eilenberg-Mac Lane spaces, Theorem 2 has the following consequence. COROLLARY 2. The surgery group Lfc+m(7TiM) Q is isomorphic to the direct sum of cohomology groups