Rigidity and Other Topological Aspects of Compact Nonpositively Curved Manifolds

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence ƒ : N —> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group Lk+m(T, w{) is isomorphic to the set of homotopy classes of k k maps [M x I rel d , GI TOP], where / is the Ac-dimensional cube (with k > 0 ). We also show that the Whitehead group Wh(r), the projective class group KQ(ZT), and the lower Kgroups K_n(ZT), n > 1 , are all isomorphic to the one element group. The higher AT-groups Kn(ZT), n > 0, are computed up to rational isomorphism type. All of these results have previously been obtained by the authors in the case that the sectional curvature values of M are strictly negative (cf. [7, 8, 9, 10]). In all the following results we let M denote a compact connected Riemannian manifold all of whose sectional curvature values are nonpositive, and we let T denote the fundamental group of M. Theorem 1. If h : TV —• M is a homotopy equivalence from a compact closed manifold N, and if dim(M) > 5, then there is a homotopy o f h to a homeomorphism. Let ^(M) denote the semisimplicial space of stable topological pseudo-isotopies of M. For any stratified fibration p : E —• B we let &(E ; p) denote the semisimplicial space of compactly supported stable topological pseudo-isotopies on E which have arbitrarily small control in B (defined in [23]). If ƒ : E -> M is a continuous map then denote by F : &>(E\p) -» &>(M) the map which is induced by ƒ . Received by the editors April 3, 1989 and, in revised form, June 27, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 18F25, 22E40, 57D50. Both authors were supported in part by the NSF. ©1990 American Mathematical Society 0273-0979/90 $1.00+ $.25 per page 59 60 F. T. FARRELL AND L. E. JONES Theorem 2. There is a stratified fibration p : E -> B, each fiber of which is a circle, and a continuous map ƒ : E —• M. The induced map F : &{E\ p) —• &(M) is a homotopy equivalence. Let ^ ( S 1 ) denote the Q-spectrum defined as follows: ^0{S ) = &>(S)\ ^(S) is the ith loop space of ^ ( S 1 ) for any / < 0 ; ^•(S) is the standard Zth delooping of ^(S) for any / > 0 (cf. [17, Appendix II]). We remark that &>(E\p) can be effectively computed from ^(S) and p : E -+ B\ for example nk{^ {E\p)) is equal to the /cth homology group of B with "stratified and twisted coefficients" ^ ( S 1 ) , where the stratification and twisting of the coefficients ^{S) is induced by that of the fibration p : E —• B (cf. [23, Appendix]). This points out the importance of understanding the fibration p : E —• 5 of Theorem 2. If M has strictly negative sectional curvature values then B of Theorem 2 is a countable infinite discrete space. (This version of Theorem 2 was proven by the authors in [8].) In general the p:E-+Bof Theorem 2 is obtained as follows. Let S(M) and RP(M) denote the unit sphere bundle of M and the real projective bundle for M. The orbits of the geodesic flow on S(M) cover the leaves of a one-dimensional foliation *& for RP(M). For each t > 0 let Et denote the union of all the closed leaves of 9 which have length less than or equal to t, and let pt : Et-+ Bt denote the quotient map obtained by collapsing each closed leaf in Et to a point. Finally let p : E —> B denote the direct limit as t —• oo of the maps pt : Et-+Bt. Let ƒ : E —• M denote the direct limit of the composite maps Et -^-> RP(M) -£2L> M. CALCULATION OF THE L-GROUPS Let I denote the /c-dimensional cube. There is a more general version of Theorem 1 which states that any homotopy equivalence h : (N, ON) —• (Mxl , Mxdl) from the compact manifold pair (N, a AT)—such that h: dN -+ M xdl is already a homeomorphism—can be homotoped to a homeomorphism modulo h\dN, provided that dim(Af) + k > 5. This more general version of Theorem 1 has as a consequence that the surgery homomorphism 0 : [ M x / r e l ^ G / T O P l ^ L ^ r , ^ ) RIGIDITY AND OTHER TOPOLOGICAL ASPECTS 61 is an isomorphism, provided that m + k > 5 and k > 0. (Here w{ : T —• Z2 is the homomorphism determined by the first Stief elWhitney class of M , m dim(M), and Lm+k(T, wx) is the (m + /c)-dimensional surgery group for F with orientation data wx which is defined in [26].) CALCULATION OF THE A^-GROUPS AND WHITEHEAD GROUPS Let R denote /c-dimensional Euclidean space. There is the following more general version of Theorem 2. Let £Pb{M x R ) denote the semisimplicial space of all stable topological pseudoisotopies on M x R which are bounded in the R -factor. Let 9h (E ; p) denote the semisimplicial space of all stable topological pseudo-isotopies on E x R which are bounded in the i?^-factor and which have arbitrary small control in B (with respect to the composite projection E x R pr0J > E -£-* B ). Then ƒ : E —• M induces a homotopy equivalence Fk : &%(E ; p) —• &b(M x R ). This more general version of Theorem 2 has the following consequences (which can be deduced from it as in [8 and 9, Appendix]): Whn(T) 0 Z(l/N) = 0, for any n > 1 and JV = [(/i + l)/2]! ; Kn(ZT) = 0 for any n < 0; K0(ZT) = 0; Kn{ZT) 0 Q = Hn(M9 Q) 0 ( 0 ^ Hn_x_Ai(M9 Q)) for any n . THE SPACE OF SELF HOMEOMORPHISMS OF M Let H(M) denote the space of self homeomorphisms of M, and let g : H (M) —• Out(T) denote the forgetful map to the outerautomorphism group of T. Note that Theorem 1 implies that g is onto. Let H0(M) denote the kernel of g. Then if m > 10 (where m = dim(M)), and if n, N are integers with 0 < n < (m 7)/3 and N = [(n + 4)/2]!, we have that Remark. The reader is referred to the following references: to [6, 9, 10, 15] for results related to Theorem 1; to [8, 21, 25] for results related to Theorem 2; to [5, 9, 10, 15] for results related to the L-group calculations; to [4, 7, 8, 14, 18, 21, 24] for results related to the AT-group and Whitehead group calculations. To obtain their results the author used in an important way results from the following references: [3, 5, 17, 19, 20, 21, 22, 23, 25, 26]. Proofs of Theorems 1 and 2. The proofs for Theorems 1 and 2 are similar in spirit to the proofs of these same theorems given in [7, 8, 9, 10] for the special case when M has strictly negative sectional 62 F. T. FARRELL AND L. E. JONES curvature values. However, there are several important technical differences, one of which we will discuss now in the context of the proof for Theorem 1. To prove Theorem 1 for M of strictly negative sectional curvature the authors use in [9, 10] a foliated control theorem for structure sets, where the foliations have the following important properties: (a) The leaves have dimension one. (b) For any a > 0 there are only finitely many leaves which have length less than a. This type of foliated control theorem unfortunately does not apply to the relevant foliations when M is allowed to have some zero sectional curvature values. (For example there are whole continuous families of closed orbits of bounded length for the geodesic flow on the unit sphere bundle of the torus T . So this foliation doesn't satisfy (b) above.) We have therefore had to extend the foliated control theory for structure sets used in [9, 10]. The following extension is proven in [12]. Let N denote a compact closed smooth manifold and let S? denote a C foliation of N by one-dimensional leaves. Let p : E —> N denote a fiber bundle over TV which has a compact closed manifold for fiber. Let h : X —• E denote a homotopy equivalence from the compact closed manifold X. We refer the reader to [12, 13] for the meaning of h being (a, e)-controlled over (N, ^) for numbers a , e > 0. See [12, Appendix] for a proof of the following theorem. Theorem 3. There is an integer k > 0 which depends only on dim(iV). Given any a > 0 there is e > 0. If the homotopy equivalence h: X —• E is (a, e)-controlled over (N9^) then there is a homotopy Ht: X x T k -+ E x T, t e [0 ,1] , of h x I : X x T —• E x T to a map H{ which is split over a triangulation of N (here 1 : T —• T is the identity map on the k-dimensional torus).