Proc. Natl. Acad. Sci. USA Vol. 86, pp. 3461-3463, May 1989 Mathematics Compact negatively curved manifolds (of dim # 3, 4) are topologically rigid (surgery/geodesic flow/homeomorphism group/marked leaves/foliated control) F. T. FARRELLt AND L. E. JONESt tDepartment of Mathematics, Columbia University, New York, NY 10027; and tDepartment of Mathematics, State University of New York, Stony Brook, NY 11794 Communicated by William Browder, January 27, 1989 (receivedfor review October 20, 1988) ABSTRACT Let M be a complete (connected) Riemannian COROLLARY 1.3. If M and N are both negatively curved, manifold having finite volume and whose sectional curvatures compact (connected) Riemannian manifolds with isomorphic lie in the interval [cl, c2d with -x < cl c2 < 0. Then any fundamental groups, then M and N are homeomorphic proper homotopy equivalence h:N -* M from a topological provided dim M 4 3 and 4. manifold N is properly homotopic to a homeomorphism, Gromov had previously shown (under the hypotheses of provided the dimension of M is >5. In particular, if M and N Corollary 1.3) that the total spaces of the tangent sphere are both compact (connected) negatively curved Riemannian bundles ofM and N are homeomorphic (even when dim M = manifolds with isomorphic fundamental groups, then M and N 3 or 4) by a homeomorphism preserving the orbits of the are homeomorphic provided dim M # 3 and 4. {If both are geodesic flows, and Cheeger had (even earlier) shown that locally symmetric, this is a consequence of Mostow's rigidity the total spaces of the two-frame bundles of M and N are theorem [Mostow, G. D. (1967) Publ. Inst. Haut. Etud. Sci. 34, homeomorphic (cf. ref. 22). It is conceivable that M and N are 53-104].} When M has infinite volume we can still calculate the even diffeomorphic; this is problem 12 ofthe list compiled by surgery L-groups of iv1M, even when dim M = 3, 4, or 5, Yau in ref. 4. All this is, of course, motivated by the fact that provided M is locally symmetric. An identification of the weak Mostow's rigidity theorem (5) implies that M and N are homotopy type ofthe homeomorphism group of (finite volume) isometrically equivalent if both are additionally assumed to M is also made through a stable range. We have previously be locally symmetric and dim M > 2. announced these results for the special case that cl = C2 = -1. Let wl:ir1M -- Z2 denote the homomorphism determined by the first Stiefel-Whitney class of M and Lj(irM) denote Let M denote a complete (connected) Riemannian manifold the j-dimensional surgery group for the fundamental group having finite volume and whose sectional curvatures lie in an r1M with orientation data wl, defined by Wall (6). Let interval [c1, c2] with -oc < c1 _ c2 < 0. We denote by M the topological manifold compactification of M constructed by O4M x Ik rel a, G/Top] + Lm+k(ITlM) [1] Gromov and Margulis (cf. ref. 1). Let N be a compact manifold such that its boundary aN decomposes as dN = a1N be the surgery homomorphism, where m = dim M (cf. refs. U a2N, where d1N, 32N are compact codimension zero 6 and 7). Because of its periodicity (in j), the following is a submanifolds of aN with calculation of Lj(ir1M) for all integers j (and dim M is not constrained). a(a1N) = a(a2N) = ajN n a2N. [0] THEOREM 2. The surgery homomorphism 0 is an isomor- phism, provided m + k > 5 and k > 0. Set AN = a(aN). Let I denote the closed interval [0, 1], and It has been shown in refs. 8 and 9 that 0 is a split for each nonnegative integer k, let Ik denote the k-fold monomorphism. cartesian product of I with itself; in particular, I° is a point. COROLLARY 2.1. The surgery group Lm+k(7rlM) 0 Q is THEOREM 1. Let h:(N, ajN, a2N, AN) -- (1A x Ik, NJ x alk, isomorphic to the direct sum of cohomology groups aM x Ik, FaM x ajk) be a homotopy equivalence of4-tuples such that h:aN -- I x alk is a homeomorphism and dim M ED H4j-k(M, aM; Q), + k > 5. Then there is a homotopy ht (of maps of4-tuples) from h = ho to a homeomorphism h1 such that the restriction where m = dim M and the cohomology groups have (un- of ht to a1N is the constant homotopy. Moreover, if the twisted) rational coefficients. restriction of h to a2N is also a homeomorphism, then we IfX is a compact manifold with possibly nonempty bound- need only assume that dim M + k > 4 and ht can be ary and A is a subspace of X, let Homeo(X, A) denote the constructed to be constant on all of aM. space of all self homeomorphisms of X that restrict to the COROLLARY 1. 1. If h:X -> M is a proper homotopy identity map onA; abbreviate Homeo(X, 0) by HomeoXand equivalence where X is a topological manifold, then h is Homeo(X x [0, 1], X x 0) by P(X)-the space of topological properly homotopic to a homeomorphism provided dim M > psuedo-isotopies on X. Let gj, g2, . denote the sequence 5. of all closed geodesics in M. (Two geodesic curves are The proof of Corollary 1.1 also uses results from refs. 1-3. considered equal if they have the same image in M.) To each COROLLARY 1.2. IfM is compact and N is aspherical (i.e., gi associate a manifold Si as follows: Si = SI x Im-1 if w1(gi) N is connected and iriN = 0 for i > 1), then M and N are = 0, where m = dim Mand S' is the circle; Si is the total space homeomorphic provided ffM and 1r1N are isomorphic, aN is of the unorientable Im-1 bundle over S1 if wl(gi) 7 0. Define empty, and dim M #& 3 and 4. (Di Homeo(Si, aSi) to be the direct limit asj co of the finite cartesian products The publication costs of this article were defrayed in part by page charge aS1) X Homeo(S2, aS2) payment. This article must therefore be hereby marked "advertisement" Homeo(Sl, in accordance with 18 U.S.C. §1734 solely to indicate this fact. x ... x Homeo(Sj, aSj). [2] Downloaded by guest on September 28, 2021 3461 3462 Mathematics: Farrell and Jones Proc. Natl. Acad. Sci. USA 86 (1989) Let Homeo(M)o denote the kernel of the natural homomor- F(M) is equipped with a foliation 9;, where the leaves of P are phism just the images of the sets X x f, f E F, under the covering projection X x F-* F(M). This foliation restricts to foliations Homeo v -k Out 7r1M. [3] 9B and 9Tfor the strata B and T, respectively. The geometry of M gives rise to additional structures, called "markings," (Recall Out 7rM is in one-to-one correspondence with the on the leaves of IB and ;IT. Each leaf of 91B is marked with homotopy classes of self homotopy equivalences of M.) an asymptotic vector field and each leaf of iTiS marked with THEOREM 3. The spaces Homeo(M)o and a geodesic. These markings give rise to the geodesicflow g' on B and a radial (imcomplete) flow r' on T as follows. (1i Homeo(S1, aSk)) x P(aM) Restricted to a leaf L of 9B, g' is the unit speed flow in the direction of the asymptotic vector field that marks L. (In fact have the same "weak homotopy type through a stable range B can be identified with the total space SM ofthe tangent unit of dimensions" provided m = dim M > 10; more precisely, sphere bundle to M and gt with the geodesic flow on SM.) there is a continuous mapfrom the second space to thefirst Restricted to a leaf L of 9T, r' is the incomplete unit speed inducing isomorphisms on frf for all integers s _ (m - 7)/3. flow described as follows. Let g be the geodesic that marks Theorem 3 is a consequence of Theorem 2 together with our L, then r' flow points in L toward g along the geodesic calculation of the stable psuedo-isotopy spaces given in ref. half-rays in L that start in g but are perpendicular to g. This 10; we combine these results using Hatcher's spectral se- g) when quence (11), stability results of Igusa (12), and work of flow is incomplete since r'(x) is defined for t _ dL(x, Burghelea and Lashof (13). Gromov showed that each com- x E L - g, and for t = 0 ifx E g. [Each leafL covers M and ponent of aM is an aspherical manifold and has a virtually thus inherits a Riemannian metric that in turn determines the nilpotent fundamental group (1). Hence, the results in refs. 2, metric dL( , ) on L.] 3, and 11-16 can be applied to rationally calculate both Consider a homotopy equivalence h:N -> M from a iijP(aM) and wj Homeo(Si, aSi) through a stable range of compact closed manifold N. Let E denote the total space of indices j. This yields the following corollary. the pullback of the bundle F(M) -- M along h, and let h*:E COROLLARY 3.1. Assume that m > 10, where m = dim M. -> F(M) be the canonical map covering h. [The bundle Then, projection F(M) -* M is induced by the natural projection of the product X x F onto its first factor.] We claim that there irs(Homeo M) 0 Q = 0 is a homotopy h* from h* = h* to a map h* that is a homeomorphism over B and over the complement of a provided 1 's _ (m - 7)/3.