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Home , Homeomorphism

Proc. Natl. Acad. Sci. USA Vol. 86, pp. 3461-3463, May 1989

Compact negatively curved (of dim # 3, 4) are topologically rigid (surgery/geodesic flow/homeomorphism /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, having finite volume and whose sectional curvatures compact (connected) Riemannian manifolds with isomorphic lie in the [cl, c2d with -x < cl c2 < 0. Then any fundamental groups, then M and N are homeomorphic proper 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 of M is >5. In particular, if M and N Corollary 1.3) that the total spaces of the tangent 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 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 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 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. of all self homeomorphisms of X that restrict to the COROLLARY 1. 1. If h:X -> M is a proper homotopy identity onA; abbreviate Homeo(X, 0) by HomeoXand equivalence where X is a , 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 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 ; Si is the total space homeomorphic provided ffM and 1r1N are isomorphic, aN is of the unorientable Im-1 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 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 Homeo v -k Out 7r1M. [3] 9B and 9Tfor the strata B and T, respectively. The 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 " 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 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 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 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. for the stratum B in F(M) and such that Remark. There are analogues to Theorems 1, 2, and 3 in h* is "blocked up" over this tubular neighborhood. To get which the assumption that "M has finite volume" is replaced such a homotopy we first flow h * over the bottom stratum by by "M is locally symmetric." These analogues require the the geodesic flow, thereby gaining (one-dimensional) foliated following changes. All homotopy equivalences of Theorem I control for h* over the bottom stratum B with respect to the must now have compact support; i.e., they must be embed- foliation of B by the flow lines of the geodesic flow. Now dings away from a large enough compact set. The set [-M x apply the structure space foliated control theorem of refs. 18 Ik rel a, G/Top] of Theorem 2 must be replaced by the set of and 19 to construct the homotopy h* over B. To get h* over compactly supported equivalence classes of compactly sup- the top stratum T [away from a tubular neighborhood for B in ported maps (and M is replaced by M). The space F(M)] we must analyze the structure ofT more carefully. Let Homeo(M)o of Theorem 3 must be replaced by the space C denote the union of all the geodesics that mark the leaves consisting of all compactly supported self homeomorphisms Of 9T; it is a submanifold called the core of T. There is a of M that are (freely) homotopic to the identity homeomor- bundle projection p :T -> C obtained as follows. For each leaf phism. L of 9T, let p:L -* g denote the orthogonal projection of L We now indicate the proof of Theorem 1. In ref. 17 we onto the geodesic g that marks L; i.e., if x E L, then outlined a proof of this result for the special case in which c1 = c2 = -1 (cf. ref. 23). The proof of the general result p(x) = r'(x), where t = dL(x, g). [4] sketched here is similar, but it contains a simplification that allows us to drop the restriction c1 = c2 = -1. There are two We now flow h* over a complement of a tubular neighbor- major ingredients: (i) a foliated control theorem for structure hood for B U C in F(M) by a (tapered) radial flow, thereby spaces, where the leaves of the foliations are either points or gainingpointwise control for h* over this complementary set. curves, and (ii) the dynamics of the geodesic flow and of a (Note B and C are disjoint.) Now apply the ordinary structure "radial (incomplete) flow." space control theorem from refs. 19 and 20 to construct h* The proofof Theorem I (when c1 = c2 = -1) outlined in ref. over the complement of a tubular neighborhood for B U C in 17 used a foliated control theorem for foliations having F(M). Finally, h* is extended over the tubular neighborhoods high-dimensional leaves. We here introduce a radial flow that of B and of C in F(M) by using the fibered version of the allows us to simplify the argument outlined in ref. 17 and, in foliated control theorem for foliations with one-dimensional particular, to avoid the use of foliated control results for leaves, also proved in ref. 19. (Ref. 19 extends Quinn's foliations having leaves of dimension >1. unfoliated fibered control results; cf. ref. 21.) Here C is To minimize the many complications that arise in the foliated by the geodesics that mark the leaves of 9#T. It is to proof, we assume that M is an odd-dimensional compact this foliation that the fibered foliated control theorem is manifold and k = 0. The geometry of M enters into the proof applied in a tubular neighborhood of C. In a tubular neigh- of Theorem I in the following way. It enables us to construct borhood of B, the fibered foliated control theorem is applied a compact stratified space F with two strata B and T, where with respect to the foliation of B by the flow lines of the B is the bottom stratum and T is the top stratum. The geodesic flow. (The core C can be identified with PM-the fundamental group F of M acts on F preserving these strata. total space of the projective m - 1 space bundle associated Let F(M), B, and T denote the orbit spaces ofX x F, X x B, to the tangent bundle of M, where m = dim M. Notice that and X x T, respectively, under the diagonal action of F, SM is a two-sheeted cover of PM and the flow lines of the where X is the universal ofM. Note that F(M) geodesic flow on SM descend to give a foliation with is a stratified space with two strata B and T. Furthermore, one-dimensional leaves for PM. It is this foliation that Downloaded by guest on September 28, 2021 Mathematics: Farrell and Jones Proc. Nati. Acad. Sci. USA 86 (1989) 3463

corresponds, under the identification of PM to C, with the 7. Kirby, R. C. & Siebenmann, L. C. (1977) Foundational Essays one described above for C using the geodesic markings.) on Topological Manifolds, Smoothings, and Triangulations Surgery procedures now allow us to "desuspend" the (Princeton Univ. Press, Princeton, NJ). 8. Farrell, F. T. & Hsiang, W.-c. (1981) Ann. Math. 113, 199-209. homotopy h* of the preceding paragraph to get a homotopy 9. Farrell, F. T. & Hsiang, W.-c. (1982) Invent. Math. 69, 155- h,:N -> M from h = ho to a homeomorphism hl. It is at this 170. step where we use the extra assumption: M has odd dimen- 10. Farrell, F. T. & Jones, L. E. (1987) Ann. Math. 126, 451-493. sion, which implies that F has index one. 11. Hatcher, A. E. (1978) Proc. Symp. Pure Math. 32, 3-21. 12. Igusa, K. (1988) K-Theory 2, 1-355. Both authors were supported by grants from the National Science 13. Burghelea, D. & Lashof, R. (1977) Ann. Math. 105, 449-472. Foundation. 14. Waldhausen, F. (1978) Proc. Symp. Pure Math. 32, 35-60. 15. Farrell, F. T. & Hsiang, W.-c. (1978) Proc. Symp. Pure Math. 1. 32, 325-337. Gromov, M. (1978) J. Differential Geom. 13, 223-230. 16. Quinn, F. (1985) Bull. Am. Math. Soc. New Ser. 12, 221-226. 2. Farrell, F. T. & Hsiang, W.-c. (1983) Am. J. Math. 105, 641- 17. Farrell, F. T. & Jones, L. E. (1988) Bull. Am. Math. Soc. New 672. Ser. 19, 277-282. 3. Farrell, F. T. & Hsiang, W.-c. (1981) J. London Math. Soc. 24, 18. Farrell, F. T. & Jones, L. E. (1988) Invent. Math. 91, 559-586. 308-324. 19. Farrell, F. T. & Jones, L. E. (1988) K-Theory 2, 401-430. 4. Yau, S.-T. (1982) Seminar on Differential Geometry (Princeton 20. Chapman, T. A. & Ferry, S. (1979) Am. J. Math. 101, 567-582. Univ. Press, Princeton, NJ). 21. Quinn, F. (1979) Ann. Math. 110, 275-331. 5. Mostow, G. D. (1967) Publ. Inst. Haut. Etud. Sci. 34, 53-104. 22. Gromov, J. (1987) Math. Sci. Res. Inst. Publ. 8, 75-263. 6. Wall, C. T. C. (1970) Surgery on Compact Manifolds (Aca- 23. Farrell, F. T. & Jones, L. E. (1989) J. Am. Math. Soc. 2, in demic, New York). press. Downloaded by guest on September 28, 2021