2 IGOR BELEGRADEK More generally, Kapovich [Kap2] proved that there is a universal unction C(?;?) such that or any incompressible singular suraces g1,

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1 INTERSECTION PAIRING IN HYPERBOLIC MANIFOLDS, VECTOR BUNDLES AND CHARACTERISTIC CLASSES Igor Belegradek November 10, 1996 Abstract. We present some constraints on the intersection pairing in hyperbolic maniolds. We also show that, given a closed negatively curved maniold M o dimension at least three, only nitely many rank k oriented vector bundles over M can admit complete hyperbolic metrics. x0. Introduction Hyperbolic maniolds are locally symmetric Riemannian maniolds o negative sectional curvature. Any complete hyperbolic maniold is a quotient o a rank one symmetric space by a discrete groups o isometries that acts reely. The main result is the ollowing 0.1. Theorem. Let Y be a nite connected CW-complex such that 1 (Y ) is torsion ree group which is not virtually nilpotent. Assume that 1 (Y ) has no nontrivial decomposition as an amalgamated product or an HNN extension over a virtually nilpotent group. Then, given a nonnegative integer n and homology classes [] 2 H m (Y ) and [] 2 H n?m (Y ), there exists K > 0 such that, (1) or any continuous map : Y! N o Y into an oriented complete hyperbolic n-maniold N that induces an isomorphism o undamental groups, and (2) or any embedding : Y! N o Y into an oriented complete hyperbolic n-maniold N that induces a monomorphism o undamental groups. the intersection number o the cycles and in N satises jh ; ij K. The assumptions on 1 (Y ) are used to invoke a compactness theorem due to Rips that the space o conjugacy classes o aithul discrete representations o 1 (Y ) into the group o isometries o a rank one symmetric space is compact. For example, these assumptions hold i Y is a closed aspherical maniold o dimension at least three where 1 (Y ) is word-hyperbolic. In particular, we get the ollowing 0.2. Corrolary. Let M be a closed oriented K(?; 1) maniold o dimension n 3 such that? is word-hyperbolic. Then there is a nite subset F H k (M) that contains the Euler class o any 1 -incompressible topological embedding o M into a complete oriented hyperbolic (n + k)-maniold. Thus, F contains the Euler class o any oriented vector bundle R k! E! M i E admits a complete F-hyperbolic structure. The corollary generalizes a result o Kapovich [Kap2] who proved that, given a closed oriented negatively curved maniold M o dimension n 3, there is a nite subset F H n (M) that contains the Euler class o any rank n vector bundle whose total space admits a complete real hyperbolic structure. Kapovich [Kap2] also proves a similar result in case M is a closed orientable surace. Namely, the Euler number o any oriented R 2 -bundle over a closed oriented surace o genus g is bounded by exp(exp(10 8 (g? 1))) provided the total space o the bundle is a complete real hyperbolic maniold Mathematics Subject Classication. Primary: 53C15, 57R20, 22E40, 20H10;. Key words and phrases. Hyperbolic maniold, vector bundle, intersection pairing, Euler class. Typeset by AMS-TEX 1

2 2 IGOR BELEGRADEK More generally, Kapovich [Kap2] proved that there is a universal unction C(?;?) such that or any incompressible singular suraces g1, g2 in a complete oriented real hyperbolic 4-maniold we have jh[ g1 ]; [ g2 ]ij C(g 1 ; g 2 ). Reznikov [Rez] showed that or any singular suraces g1, g2 in a closed oriented negatively curved 4-maniold M with the sectional curvature pinched between?k 2 and?k 2 there is a bound jh[ g1 ]; [ g2 ]ij C(g 1 ; g 2 ; k; K; (M)) or some universal unction C. Some vector bundles appear naturally as hyperbolic maniolds. For example, given a real hyperbolic maniold M the maniold M R k admits a real hyperbolic structure while the tangent bundle T M admits a complex hyperbolic structure. However, aside rom these trivial cases, it is not easy to construct hyperbolic structures on vector bundles. All known examples are based on some subtle geometric constructions. An orientable R 2 -bundle over a closed oriented surace o genus g is known to admit a real hyperbolic structure provided the Euler class e o the bundle satises jej < g [Luo1,2] (c. [GLT], [Kap1], [Kui1,2]). For nonorientable bundles over nonorientable suraces the condition jej [ g ] implies the existance o a real hyperbolic 8 structure [Bel1]. Complex hyperbolic structures exist on orientable R 2 -bundles over closed oriented suraces provided je + 2g? 2j < g [GKL]. There are examples o real hyperbolic structures on nontrivial orientable R 2 -bundles over closed real hyperbolic 3-maniolds [Bel2]. For any stable characteristic class c (e.g. the total Pontrjagin class) there is a uniorm bound on the number o elements o the ring H (M; Z) that can be represented as the class c o the normal bundle o an immersion o M into a (G; X)-maniold where G is real algebraic or semisimple with nite center. In act, the bound is the number o connected components o the space Hom( 1 (M); G). Note that a vector bundle is determined up to a nite number o possibilities by its rational Pontrjagin classes and the rational Euler class. In particular, we get the ollowing Theorem. Let M be a closed smooth maniold and k be a positive integer. Assume that either k is odd or k > dim(m). Then the set o isomorphism classes o the normal bundles o smooth immersions o M into (possibly incomplete) hyperbolic (n + k)-maniolds is nite. Combining 0:2 and 0:3 we get 0.4. Theorem. Let M be a closed oriented negatively curved n-maniold with n 2 and k be a positive integer with n+k 5. Then the set o isomorphism classes o the normal bundles o smooth 1 -incompressible embeddings o M into a complete hyperbolic (n + k)-maniold is nite. In particular, the total spaces o only nitely many smooth oriented rank k vector bundles over M can admit complete hyperbolic structures. Note that according to a result o Anderson [A] the total space o any vector bundle over a closed negatively curved maniold admits a complete metric with the sectional curvature pinched between two negative constants. Finally, I would like to thank W. P. Goldman, M. Kapovich, J. M. Rosenberg, and C. Rourke or many helpul discussions. x1. Varieties o representations and vector bundles Let B be a nite connected CW -complex with the universal cover B. ~ Let G be a Lie group that acts eectively and analytically on a contractible n-maniold X. Given a homomorphism : 1 (B)! G, the group 1 (B) acts on B ~ X via (b; x) = (b; ()x). The quotient map ( B ~ X)= 1 (B) = E! B=1 ~ (B) = B is a at (G; X)-bundle with ber X and holonomy. Isomorphism classes o at (G; X)-bundles are in one-to-one correspondence with the space o conjugacy classes o representations Hom( 1 (B); G)=G. The vector bundle B ~ T X! B ~ X decsendes to a vector bundle over E. The bundle is "vertical" in the sense that its bers are tangent to the bers o E. Clearly, isomorphic (G; X)-bundles (disregarding the at structure) have isomorphic vertical bundles. Since X is contractible, the bundle E has a section s which is unique up to homotopy. For homotopic sections s 2 and s 1, the bundles s and 1 s 2 are isomorphic, hence, the isomorphism class

3 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES 3 o s is determined by. In act, the isomorphism class o s only depends on the conjugacy class o in Hom( 1 (B); G) since isomorphic (G; X)-bundles have isomorphic vertical bundles. Moreover, by the covering homotopy theorem, i representations 1 and 2 lie in the same connected component o the representation variety Hom( 1 (B); G), the bundles E 1 and E 2 are isomorphic as (G; X)-bundles. Thereore, the isomorphism class o an (G; X)-bundle E (and, hence, the isomorphism class o its vertical vector bundle) depends only on the connected component o in Hom( 1 (B); G). Thus we have constructed a locally-constant map V : Hom( 1 (B); G)! V ect n (B) that takes a representation rom Hom( 1 (B); G) to the isomorphism class o the corresponding vertical bundle. Furthermore, given a characteristic class c : V ect n (B)! H (B) we get a locallyconstant map c : Hom( 1 (B); G)! H (B). In case the group G is real algebraic, complex algebraic, or semisimple with nite center, Hom( 1 (B); G) has nitely many connected components [G1], in particular, the map V has nite image Proposition. Let be a continuous map o B into an (G; X)-maniold N with holonomy h : 1 (N)! G. Then the pullback T N o the tangent bundle to N via is isomorphic to the pullback o the vertical bundle o the (G; X)-bundle E via a section, where = h : 1 (B)! G. Proo. Let be a continuous map o B into an (G; X)-maniold N with holonomy homomorphism h : 1 (N)! G. Lit to an equivariant map o universal coverings ~ : ~ B! ~ N and compose with the developing map d : ~ N! X. Thus, we get an equivariant map d ~ : ~ B! X. The group 1 (B) acts via by bundle isomorphisms on T ~ N, and 1 (B) acts via h by bundle isomorphisms on T X. Since d is an equivariant local dieomorhism, we have an equivariant isomorphism o pullback bundles ~ (T ~ N) = ~ d (T X). Note that the bundle ~ d (T X) is equivariantly isomorphic to the pullback o the bundle ~ B T X! ~ B X via the map ~ F = (id; d ~ ) : ~ B! ~ B X. The group 1 (B) acts on ~ B T X by bundle isomorphisms, so ~ B T X descendes to the vertical bundle over E. Since ~ F is equivariant, 1 (B) also acts on ~ F ( ~ B X) by bundle isomorphisms and, thereore, the bundle ~ F ( ~ B X) descendes to F, the pullback o the vertical bundle via the section F. Similarly, the bundle ~ (T ~ N) descendes to T N. Thus the equivariant isomorphism ~ (T ~ N) = ~ d (T X) = ~ F ( ~ B X) induces the bundle isomorphism T N = F Corollary. Assume the space Hom( 1 (B); G) has d < 1 connected components. Then there exist d real vector bundles over B such that, or any continuous map o B into an (G; X)-maniold N, the pullback bundle T N is isomorphic to one o the bundles. We denote [E] the stable equivalence class o a vector bundle E in the real K-theory g KO(B). Notice that, i B is a closed smooth maniold and is a smooth immersion, the normal bundle o the immersion has the property T B = T N; thereore, in this case [ ] = [ T N]? [T B]. Characteristic classes are natural transormations o contravariant unctors VectR(?)! H (?; ). A characteristic class is called stable i descendes to a natural transormation g KO(?)! H (?; ). For example, the total Pontrjagin class is stable Corollary. Let B is a closed smooth maniold and the space Hom( 1 (B); G) has d < 1 connected components. Then there exists a nite subset o g KO(B) o size at most d that contains the equivalence class o the normal bundle o any immersion o B into an (G; X)-maniold N. In particular, or any stable characteristic class c : V ect(b)! H (B; Z), there is a nite subset o H (B; Z) o size at most d that contains the c-image o the normal bundle o any immersion o B into an (G; X)-maniold N. Proo. Let i : B! N be an immerion with the normal bundle i. Then i T B = i T N. Passing to real K-theory we get [ i ] = [i T N]? [T B]. By 1.2, there are at most d possibilities or the isomorphism class o i T N while T B is a constant.

4 4 IGOR BELEGRADEK 1.4. Remark. In case the G-action preserves an almost complex structure on X, any (G; X)- maniold is almost complex. In paricular, the vertical and tangent bundles are complex and 1.3 holds or almost complex immersions (o course KO must be replaced with KU). I am most grateul to J. M. Rosenberg or explaining to me the ollowing nice act Proposition. Let Y be a nite CW-complex and k be a positive integer. Then the set o isomorphism classes o oriented real (complex, respectively) rank k vector bundles over Y with the same rational Pontrjagin classes and the rational Euler class (rational Chern classes, respectively) is nite. Proo. To simpliy notations we only give a proo or complex vector bundles and then indicate necessary modications or real vector bundles. We need to show that the \Chern classes map" (c 1 ; : : : ; c k ) : [Y; BU(k)]! H (Y; Q) is nite-to-one. First, notice that c 1 classies the line bundles (c. 2.8), so we can assume k > 1. The integral ith Chern class c i 2 H (BU(k); Z) = [BU(k); K(Z; 4i)] can be represented by a continuous map i : BU(k)! K(Z; 2i) such that c i = i ( 2i) where 2i is the undamental class o K(Z; 2i). (Recall that, by Hurewicz theorem H n?1 (K(Z; n); Z) = 0 and H n (K(Z; n); Z) = Z; the class in H n (K(Z; n); Z) = Hom(H n (K(Z; n); Z); Z) corresponding to the identity homomorphism is called the undamental class and is denoted n.) It denes \Chern classes map" c = ( 1 ; : : : ; k ) : BU(k)! K(Z; 2) K(Z; 2k) = : We now check that the map induces an isomorphism on rational cohomology. According to [GM, 7.5, 7.6] or even n, H n (K(Z; n); Q) = Q[ n ]. By Kunnet ormula H ( k i=1k(z; 2i); Q) = k i=1h (K(Z; 2i); Q) = k i=1q[ 2i ] = Q[ 2 ; : : : 2k ] and under this ring isomorphism s1 2 s k 2k corresponds to s1 : : : 2 s k 2k. It is well known that H (BU(k); Q) = Q[c 1 ; : : : ; c k ] [MS, 14.5]. Thus the homomorphism c : H ( k i=1k(z; 2i); Q))! H (BU(k); Q) denes a homomorphism Q[ 2 ; : : : 2k ] = Q[c 1 ; : : : ; c k ] that takes 2i to c i. Since this is an isomorphism, so is c as promised. Thereore c induces an isomorphism on rational homology. Then, since BU(k) is simply connected, the map c must be a rational homotopy equivalence [GM, 7.7]. In other words the homotopy theoretic ber F c o the map c has nite homotopy groups. Consider an oriented rank k vector bundle : Y! BU(k) with characteristic classes c : Y!. Our goal is to show that the map c has at most nitely many nonhomotopic litings to BU(k). Look at the set o litings o c to BU(k) and try to construct homotopies skeleton by skeleton using the obstruction theory. The obstructions lie in the groups o cellular cochains o Y with coecients in the homotopy groups o F c. (Note that the bration F c! BU(k)! has simply connected base and ber (since k > 1), so the coecients are not twisted.) Since the homotopy groups o F c are nite, there are at most nitely many nonhomotopic litings. This completes the proo or complex vector bundles. For oriented real vector bundles o odd rank the same argument works with c = (p 1 ; : : : ; p [k=2] ), where p i is the ith Pontrjagin class. Similarly, or oriented real vector bundles o even rank we set c = (e; p 1 ; : : : ; p k=2?1 ), where e is the Euler class. We point out as a warning that the argument given above ails or BO(k), due to the act BO(k) is not simply connected (i.e. the map C = (p 1 ; : : : ; p [k=2] ) o BO(k) to the product o Eilenberg- MacLane spaces is not a rational homotopy equivalence even though it induces an isomorphism on rational cohomology). Probably the conclusion o the theorem is true i we use characteristic classes with twisted coecients. The need or something like "twisted Euler class" can be seen rom the computation i (BSO(k)) = i (BO(k)) or i 2; thus the image o the usual "Euler class" survives in k (BO(k)). We choose to avoid dealing with twisted classes by considering oriented bundles Remark. Note that, in case either k is odd or k > dim(y ), the rational Euler class is zero, and, hence rational Pontrjagin classes determine the vector bundle up to a nite number o possibilities.

5 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES Corollary. Let B is a closed smooth oriented maniold and the space Hom( 1 (B); G) has d < 1 connected components. Then the ollowing holds: (1) Set k = dim(x)? dim(b). I either k is odd or k > dim(b), then the set o isomorphism classes o the normal bundles o smooth immersion o B into an (G; X)-maniold N is nite. (2) I the G-action preserves an almost complex structure on X, then the set o isomorphism classes o the (complex) normal bundles o almost complex immersion o B into an (G; X)-maniold N is nite. x2: Invariants o continuous maps Throughout the paper we assume that all maniolds are Hausdor, paracompact and connected Denition. Let B be a topological space and X B be a set that may depend on B. Let be a map that, given an oriented maniold N and a continuous map rom B into N, produces an element o X B. We call an invariant i the two ollowing conditions hold: (1) Homotopic maps 1 : B! N and 2 : B! N have the same invariant. (2) Let h : N! L be an orientation-preserving homeomorphism o N onto an open subset o L. Then, or any continuous map : B! N, the maps : B! N and h : B! L have the same invariant Example: tangent bundle. Assume B is paracompact and X B is the semigroup VectR(B) o real vector bundles over B. Given a continuous map : B! N, set ( : B! N) = T N, the pullback o the tangent bundle to N under. Then is an invariant. More generally, let be a unctor rom the category o maniolds and open embeddings to the category o vector bundles over the maniolds. Then ( : B! N) = N is an invariant. For instance, one can take N to be an iterated tangent bundle, or their Whitney sums, tensor, or exterior products. Furthermore, stable equivalence class o N and the characteristic classes o N are invariants Example: normal bundles. Assume B is a closed smooth maniold and X B = KO(B). Given a continuous map : B! N, set ( : B! N) = [ T N]? [T B]. Notice that, i is a smooth immersion, the normal bundle o the immersion has the property T B = T N; thereore, in this case [ ] = [ T N]?[T B]. Clearly, any stable characteristic class o [ T N]?[T B] is an invariant with values in X B = H (B; ). In case each N is almost complex all bundles are complex and KO should be replaced with KU Euler class. Since the Euler class is non-stable we need a dierent approach to dene the Euler class o a continuous map. Assume B is a compact topological space and N is an oriented n-maniold. All (co)homology groups in this section have integer coecients. Set X B = Hom(H (B); H (B)). Let : B! N be a continuous map. For every positive integer m we dene a group homomorphism " m () o H m (B) into H n?m (B) as the composition o the ollowing homomorthisms H m (B)????! Hm (N; Nn(B)) = H n?m ((B))????! H n?m (B) where the middle isomorphism is the Poincare Duality (see [D, VIII.7.2]). Notice that we do not have to use Cech cohomology because the natural transormation H n?m ((B))! H n?m ((B)) is an isomorphism since (B) is ENR [D, VIII.6.12] (recall that in a Hausdor paracompact maniold any compact subset is ENR [D, VIII.1.3, IV.8.10, IV.8.11]). We denote "() the collection o homomorphism " m ()g. We next show that "() is an invariant. 2.5 Intersections. There is an obvious connection between "() and the intersection pairing in N. Let 2 H m (B) and 2 H n?m (B) where n = dim(n). The intersection o and in N can be computed as ollows. Since B is compact one can nd a compact K with (B) K N.

6 6 IGOR BELEGRADEK Thus we get the ollowing commutative diagram. H m (B)????! Hm (N; Nn(B)) D?1????! H n?m ((B))??y i x?? x??i H m (K)????! Hm (N; NnK)????! D?1 H n?m (K)????! H n?m (B)????! H n?m (B) Let A 2 H (K) and B 2 H (K) be the Poincare duals o ; 2 H (K) respectively. Then the intersection number is given by I( ; ) = (?1) m(n?m) < A; >= (?1) m(n?m) < B; >. It is easy to check that I( ; ) does not depend on K. Furthermore, (?1) m(n?m) I( ; ) =< A; >=< "(); >=< B; >=< "(); > : 2.6. Denition. Let B be a closed oriented m-maniold. We dene the Euler class e() o a continuous map : B! N to be " m ()-image o the undamental class [B] o B. Thus, e() depends on the choice o orientation on B. Recall that the Euler class o a smooth embedding : M! N is the Euler class o the normal bundle o (M) in N. I is a topological embedding, one can still dene its Euler class [D, VIII.11.10]. Notice that our denition generalizes the one in [D]. Indeed, according to [D, VIII.11.18] the Euler class N M dened in [D] has the property N M \ [N; Nn(M)] = [M] where [N; Nn(M)] is the undamental class o the pair (N; Nn(M)). Our denition o the Euler class has been o some use in topology (see e.g. [CS p137]). However no careul treatment is available Remark. In case n? m is odd and : M! N is homotopic to an embedding one can prove that e() has order two in H n?m (M). Indeed, it is enough to check the assumptions o VIII in [D]. As remarked ater VIII o [D], the assumptions are always satised i M and N are ENR, which is the case since M and N are paracompact, Hausdor maniolds. Moreover i n? m = 1, then e() = 0 because H 1 (M) is torsion ree by the universal coecients theorem Remark. For vector bundles, the Euler class is the primary obstruction to existence o an everywhere nonzero section. Moreover, or 2-plane bundles, the Euler class turns out to be the only invariant, i.e. orientable 2-plane bundles over a Hausdor paracompact space X are isomorphic i and only i their Euler classes are equal. In addition, every element o H 2 (X) can be realized as the Euler class o some orientable 2-plane bundle [H, I.3.8, I.4.3.1] Euler class is an invariant. Let L be a compact subset o N such that (K) L and let U be an open subset o N with L U. The ollowing diagram commutes by naturality o the cap product, H m (K)????! Hm (N; Nn(K)) D?1????! H n?m ((K)) x i?? x??i H m (K)????! Hm (U; UnL) D?1????! H n?m (L)????! H n?m (K)????! H n?m (K) thus, the homomorphism " m () can be computed using the lower raw o the diagram. In other words we can replace K by a larger compact set and N by a smaller open subset. The property (1) in the denition o an invariant says that, or homotopic maps 1 ; 2 : K! N, we have " m ( 1 ) = " m ( 2 ). Indeed, choose a homotopy F : [0; 1]K! N and put L = Im(F ). Then both " m ( 1 ) and " m ( 2 ) can be computed as the compositions o the ollowing homomorphisms. H m (K)????! H m (N; NnL) = H n?m (L))????! H n?m (K)

7 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES 7 Now since the maps 1 ; 2 : K! L are homotopic, " m ( 1 ) = " m ( 2 ). The property (2) in the denition o an invariant ollows rom the observation below. Let 1 ; 2 : K! U be continuous maps and let g be a sel-homeomorphism o U such that g 1 = 2. Then " m ( 1 ) = " m ( 2 )deg(g) as the diagram below shows. H m (K) 1????! Hm (N; Nn 1 (K)) D?1????! H n?m ( 1 (K))? g?y x??g H m (K) 2????! Hm (U; Un 2 (K)) D?1????! H n?m ( 2 (K)) 1????! H n?m (K) 2????! H n?m (K) Remark. Invariants o a continuous map : B! N can be computed in any open neighborhood o (B) thanks to the property (2). Let B be a nite connected CW-complex and : B! N be an embedding (i.e. homeomorphism onto its image). Let p : N ~! N be a covering map with p 1 ( N) ~ = 1 (B). One can lit to an embedding ~ : B! N. ~ Thus the map pj ~! (B) is a homeomorhism. Using compactness o (B) B, one can nd an open neighborhood U ~ o (B) ~ such that pj ~U! U is a homeomorphism. Since invariants ( ) ~ and () can be computed in U ~ and U repectively and p : U ~! U is an orientation-preserving homeomorphism (with respect to the induced orientation on N) ~ we conclude ( ) ~ = (). x3. Invariants o discrete representations Any invariant o continuous maps gives rise to an invariant o discrete representations as ollows. Let Y be a nite connected CW-complex and let X be a contractible n-maniold without boundary (thus, Y is compact and X is orientable). Start with a discrete representation : 1 (Y; y)! Homeo + (X) such that the group? = ( 1 (Y; y)) acts reely and properly on X. Thus p : X! N = X=? is a covering. By choosing a point x 2 X we x an isomorphism :? = 1 (N; p(x)). The homomorphism : 1 (Y; y)! 1 (X; p(x)) denes a homotopy class o maps [] 2 [(Y; y); (N; p(x))] [Sp, 8.11, Thm 11]. It is easy to see the ree homotopy class [] 2 [Y ; N] is independent o the choice o x (see Lemma 3.2 or the proo). Given an invariant o continuous maps, we dene the associated invariant o discrete representations as () = (). In particular, the Euler class o a discrete representation is dened as e() = e(). Since () depends only on the ree homotopy class o, () and e() are well dened Lemma. The ree homotopy class [] 2 [Y ; N] is independent o the choice o a point x 2 X Proo. Choosing a point x k 2 X, k 2 1; 2g we x an isomorphism k :? = 1 (N; p(x k )). The homomorphism k : 1 (Y; y)! 1 (N; p(x k )) denes a homotopy class o maps [ k ] 2 [(Y; y); (N; p(x k ))] [Sp, 8.11, Thm 11]. Similarly, the isomorphism 2?1 1 : 1 (N; p(x 1 ))! 1 (N; p(x 2 )) yields a unique homotopy class o maps [h] 2 [N; p(x 1 ); N; p(x 2 ]. Notice that ( 2?1 ) ( 1 1 ) = 2, hence [h 1 ] = [ 2 ]. It remains to show that h is reely homotopic to the identity. Let ~! be a path in X joining ~!(0) = x 1 and ~!(1) = x 2 ; denote! = p(~!). According to [Sp 7.3.2], there is a map : (N; p(x 1 ))! (N; p(x 2 ))!-homotopic to the identity (that is, there is a homotopy F t o the identity and such that F t (p(x 1 )) =!(t)). We are to show that and h are homotopic. Let be an arbitrary loop at p(x 1 ) =!(0), then the loop () is homotopic to the loop!!?1. (Indeed, put! t =!j [0;t] ; then! t?1 F t ()! t is an!(0)-preserving homotopy o and!?1 ()!). In act, since is an isomorphism, any element o 1 (N;!(1)) is o the orm [()] = [!!?1 ]. Lit to a (unique) path ~ : [0; 1]! X with (0) = x 1, so (1) = (x 1 ) where 2? such that 1 () = []. Consider the path (~!) ~ ~!?1 joining ~!(1) = x 2 to (x 2 ); the path is a lit o!!?1. Thus 2 () = [!!?1 ] = ([]) = 1 (), that is, = 2?1. Thereore, 1 [] = [h] in [N; p(x 1 ); N; p(x 2 )] (see [Sp, 8.11, Thm 11]). Since is reely homotopic to the identity, we are done.

8 8 IGOR BELEGRADEK 3.3. Claim. Given k 2 1; 2g, let k : 1 (Y; y)! Homeo + (X) be a discrete representation such that the group k ( 1 (Y; y)) acts reely and properly on X and. Assume 1 and 2 are conjugate by an orientation-preserving homeomorphism ~g o X. Then ( 1 ) = ( 2 ) Proo. Denote? k = k ( 1 (Y; y)). Let :? 1!? 2 be the isomorphism induced by ~g, so that 1 = 2, and p k : X! X=? k = N k be the natural projection. Choose a point x 1 2 X and put x 2 = ~g(x 1 ) This xes isomorphisms k :? k = 1 (N k ; p k (x k )). The homomorphism k k : 1 (Y; y)! 1 (N k ; p k (x k )) denes a homotopy class o maps [ k ] 2 [(Y; y); (N k ; p k (x k ))] [Sp, 8.11, Thm 11]. The isomorphism 2?1 1 : 1 (N; p(x 1 ))! 1 (N; p(x 2 )) is induced by a homeomorphism [g] 2 [N; p(x 1 ); N; p(x 2 )], where ~g is the lit o g. Notice that ( 2?1 1 ) ( 1 1 ) = 2 1 = 2 2 hence [g 1 ] = [ 2 ]. Sinve is an invariant, ( 1 ) = ( 2 ) as desired Remark. Let G be a Lie group that acts smoothly on a contractible maniold X. In x 1 we constructed a locally-constant map V : Hom( 1 (Y ); G)! V ect n (Y ) that takes a representation rom Hom( 1 (Y ); G) to the isomorphism class o the corresponding vertical bundle. Set D( 1 (Y )) = 2 Hom( 1 (Y ); G) : ( 1 (Y )) acts reely and properly discontinuously on Xg: Let : D( 1 (Y ))! V ect n (Y ) be an invariant o discrete representations associated with the invariant o maps dened in 2.2. Recall that is dened as ollows. Take a continuous map : Y! N = X=( 1 (Y )) that induces ; then () is the isomorphism class o the bundle T N. Then, according to the proposition 1.1, is the restriction o V. Thereore, the stable equivalence class o T N and characterisic classes o T N are also restrictions o the corresponding invariants dened on the whole space Hom( 1 (Y ); G). However, in general, the Euler class is not a restriction o an invariant coming rom the map V. x4: Geometric convergence In this section we discuss some basic acts on geometric convergence. The notion o geometric convergence was introduced by Chaubaty (see [Bou]). More details relevant to our exposition can be ound in [CEG], [Lok] and [BP]. In this section we let G be a Lie group equiped with some let invariant Riemmanian metric Denition. Let C(G) be the set o all closed subgroups o G. Dene a topology on C(G) as ollows. We say that a sequence? n g 2 C(G) converges to? geo 2 C(G) geometrically i the ollowing two conditions hold: (1) I n k 2? nk converges to 2 G, then 2? geo ; (2) I 2? geo, then there is a sequence n 2? n with n! in G Fact. [CEG] The space C(G) is compact and metrizable Fact. [Lok]? n!? geo i or every compact subset K G the sequence? n \ K!? geo \ K in the Hausdor topology (i.e. or any " > 0, there is N such that, i n > N, then? n \ K lies in the "-neihgborhood o? geo \ K and? geo \ K lies in the "-neihgborhood o? n \ K) Fact. [Lok] Let? geo G is a discrete subgroup. Then there is " > 0 such that, or any? n!? geo in C(G) and any compact K G, there is N such that, i n > N and 2? geo \ K, then there is a unique n 2? n \ K that is "-close to. In particular,? n is discrete or n > N, since e 2? n is the only element o? n, that is in the "-neighbourhood o e 2 G.

9 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES Lemma. Let G be a Lie group and? n be a sequence o discrete groups that converges geometrically to? =? geo. I the identity component? 0 o? geo is compact, the sequence? n g has unbounded torsion (i.e. or any positive integer N, there is 2? n(n) o nite order which is greater than N). Proo. Choose > 0 so small that the closed 4-neighborhood U o? 0 is disjoint rom? n? 0 (this is possible since? 0 is compact). Given M > 1, or large n,? n and? are e=m-hausdor close on the compact set U. Take an arbitrary element g 2? 0 in -neighborhood o the identity e and choose g n 2? n so that it is e=m-close to g. First, show that g n has nite order. Suppose not. Since U is compact and < g n > is discrete and innite, there is a smallest k > 1 with (g n ) k =2 U. So, (g n ) k?1 2 U. The metric is let invariant, hence, d((g n ) k?1 ; (g n ) k ) = d(g n ; e) < d(g; e) + d(g n ; g) < + =M < 2: Since? n and? are e=m-hausdor close on U and since?\u =? 0, we get d((g n ) k?1 ;? 0 ) < =M <. This implies d((g n ) k ;? 0 ) < 3 and, thus, (g n ) k 2 U, a contradiction. Thus g n has nite order. We have just showed that any neighborhood o g contains an element o nite order g n 2? n (or n large enough). So we get a sequence g n converging to g. Assume the torsion o? n is uniormly bounded by N. Then, or any nite order element h 2? n, we have h N! = e. In particular e = (g n ) N! converges to g N!. We, thus, proved that any element g in? 0 that is -close to the identity satises g N! = e. This is absurd. (Indeed, it would mean that the map? 0!? 0 that takes g to g N! maps the -neighborhood o the identity to the identity. But the map is clearly a dieomorphism on some neighborhood o the identity). Thus,? n has unbounded torsion as desired Remark. Let? n be a sequence o torsion ree groups converging to a discrete group? geo in C(G). Then? geo is torsion ree. Indeed, choose 2? geo with k = e. Find a sequence n!. Then n k! e. By 4.4 we have n k = e or large n. Since? n are torsion ree, k = 1 as desired Lemma. Let G be a Lie group with the identity component G 0 and? n be a sequence o discrete groups that converges geometrically to? =? geo. Then the identity component? 0 o? geo is nilpotent. Proo. The group? geo is closed, so it is a Lie group. So the identity component? 0 o? geo is a Lie group; we are to show that? 0 G 0 is nilpotent. Let U be a neighborhood o the identity that lies in a Zassenhaus neighbourhood in G 0 (see [Rag, 8.16]). Being a Lie group? 0 is generated by any neigborhood o the identity, in particular by V = U \? 0. To show? 0 is nilpotent, it suces to check that, or some k, V (k) = eg, that is, any iterated commutator o weight k with entries in V is trivial [Rag 8.17]. Fix an iterated commutator [v 1 : : : [v k?2 [v k?1 ; v k ]] : : : ] with v k 2 V and choose a sequences gk n! v k where gk n 2 n(). For n large enough, the elements gk n; : : : gn k lie in a Zassenhaus neighbourhood. Then by Zassenhaus-Kazhdan-Margulis lemma [Rag 8.16] the group < gk n; : : : gn k > lies in a connected nilpotent Lie subgroup o G 0. The class o any connected nilpotent Lie subgroup o G 0 is bounded by dim(g 0 ). Thus, or k > dim(g 0 ), the k-iterated commutator [g n : : : 1 [gn k?2 [gn ; k?1 gn k ]] : : : ] is trivial, or large n. This implies [v 1 : : : [v k?2 [v k?1 ; v k ]] : : : ] is trivial and, thereore,? 0 is nilpotent Theorem. Let G be a Lie group with the identity component G 0 and be a nitely generated group without nontrivial normal nilpotent subgroups. Let k 2 Hom(; G) be a sequence o discrete aithul representations that converges algebraically to a representation. Then is aithul. Proo. It suces to prove that the group K = Ker() is nilpotent. Let V be a set o generators or K (maybe innite). To show K is nilpotent, it suces to check that, or some k, V (k) = eg, that is, any iterated commutator o weight k with entries in V is trivial [Rag 8.17]. Fix an iterated commutator [v 1 : : : [v k?2 [v k?1 ; v k ]] : : : ] with v k 2 V. For n large enough, the elements n (v 1 ); : : : ; n (v k ) lie in a Zassenhaus neighbourhood o G 0. Then by Zassenhaus-Kazhdan-Margulis lemma [Rag 8.16] the group < n (v 1 ); : : : ; n (v k ) > lies in

10 10 IGOR BELEGRADEK a connected nilpotent Lie subgroup o G 0. The class o any nilpotent subgroup o G 0 is bounded by dim(g 0 ). Thus, or k > dim(g 0 ), the k-iterated commutator [ n (v 1 ) : : : [ n (v k?2 )[ n (v k?1 ); n (v k )]] : : : ] is trivial, or large n. Since n is aithul [v 1 : : : [v k?2 [v k?1 ; v k ]] : : : ] is trivial and we are done Theorem. [Lok] Let X be a simply connected homogeneous Riemannian maniold and G be a transitive group o isometries. Let? n be a sequence o torsion ree subgroups o G converging geometrically to a discrete subgroup? geo. Let U X be a relatively compact open set. Then, i n is large enough, there exists a relatively compact open set V with U V and smooth embeddings ~' n : U! V such that ~' n cover embeddings ' n : U=? geo! V=? n p U? y ~'n????! V U=? geo????! 'n? y pn V=? n The embeddings ~' n converges C r -uniormly to the inclusion. Sketch o the Proo. Let K = g 2 G : g(v ) \ V 6= ;g, so K is compact. By 4.4, i we take n suciently large, then or any 2? geo \ K there exists a unique n 2? n \ K that is close to. It denes a bijective map r n rom? geo \ K onto? n \ K. Our goal is to construct embeddings ' n : U! V that are close to the inclusion and r n -equivariant (in the sense that x = (y) or 2? geo i ' n (x) = r n ()' n (x)) The construction is non-trivial and we reer to [Lok] or a complete proo Let be a nitely generated group and H n F be the rank one negatively curved symmetric space over F (i.e. H n F is a real, complex or quaternion hyperbolic n-space or the Cayley plane). Let G = Isom(H n F ); we equip Hom(; G) with algebraic (i.e. pointwise convergence) topology Theorem. Let k 2 Hom(; G) be a sequence o representations converging algebraically to a representation where, or every k, the group k () is discrete and the sequence k ()g has uniormly bounded torsion. Suppose that () is an innite group without nontrivial normal nilpotent subgroups. Let n k() be a subsequence o k () that converges geometrically to? geo (such a subsequence always exists since C(G) is compact). Then? geo is discrete (and so is ()? geo ). Proo. Being a closed subgroup? geo is a Lie group. By the lemma 4.7 the identity component? 0 o? geo is nilpotent Lie group. Since () does not have a nontrivial normal nilpotent subgroup, () \? 0 = eg, so () is discrete. Using the Selberg lemma we nd a torsion ree subgroup () o nite index. The group is innite since () is. Notice that the group may have at most two xed points at innity (any isometry that xes at least three points is elliptic [BGS, p84]). Suppose the identity component? 0 o? geo is nontrivial. Lemma. The group is virtually nilpotent. Proo. According to [Bow1, 3.3.1], the nilpotence o? 0 implies one the ollowing mutually exclusive conclusions: Case 1.? 0 has a xed point in H n F (hence? 0 is compact which is is impossible by 4.5). Case 2. The xed point set o? 0 is a point p n F. Since normalizes? 0 (in act,? 0 is a normal subgroup o? geo ) it has to x p too. I p is the only xed point o, then every nontrivial element o is parabolic [EO, 8.9P] Any parabolic preserves horospheres centered at p [BGS, p84], thereore, according to [Bow2] is virtually nilpotent.

11 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES 11 I xes two points at innity, then it acts reely and properly discontinuously on the geodesic joining the points. Hence = Z, the undamental group o a circle. Case 3.? 0 has no xed points in H n F and preserves setwise some bi-innite geodesic. In this case the xed point set o? 0 is the endpoints p and q o the geodesic. Indeed,? 0 xes each o the endpoints, since? 0 is connected. Assume? 0 xes some other point n F. Then any element o? 0 is elliptic ([BGS, p85]) and, as such, it xes the bi-innite geodesic pointwise. Thus? 0 has a xed point in H n F which contradicts the assumptions o the Case 3. Since normalizes? 0, it leaves the set p; qg invariant. Moreover, preserves p; qg pointwise because contains no elliptics (any isometry that ips p and q has a xed point on the geodesic joining p and q). Thereore, = Z as beore. Hence, is virtually nilpotent as desired. Thus, () has a nilpotent subgroup o nite index and, thereore, has a normal nilpotent subgroup o nite index. This contradicts the assumption that () is an innite group without nontrivial normal nilpotent subgroups Remark. Let? be a nitely generated discrete subgroup o Isom(H n F ) that has a normal nilpotent subgroup. Then? is, in act, virtually nilpotent. Indeed, by Selberg lemma we can assume? is torsion ree. Then, repeating the arguments above, we see that? x a point at innity and, hence, is virtually nilpotent. Conversely, any virtually nilpotent group clearly has a normal nilpotent subgroup o nite index Corrolary. Let be a nitely generated discrete subgroup o G = Isom(H n F ) that is not virtually nilpotent. Then the set o aithul discrete representations o into G is a closed subset o Hom(; G). Proo. Let a sequence n o aithul discrete representations converge to 2 Hom(; G). According to 4.12 the group has no normal nilpotent subgroup. Then 4.8 implies is aithul. By compactness n () has a subsequence that converges geometrically to? geo which is a discrete group by Thereore, ()? geo is also discrete as wanted. x5: A compactness theorem 5.1. In this section we state a compactness theorem due to Rips (see [BF] and the review o the paper by Paulin in MR96h:20056). Earlier versions o the theorem have been proved by Thurston, Morgan and Shalen [Mor]. We are grateul to M. Bestvina or useul communications on the Rips' theorem. Assume that G = Isom + (H n F ), the group o all orientation-preserving isometries o Hn F. Let F D(; G) Hom(; G) be a subspace o aithul representations such that the groups n ((Y; y)) are discrete Denition. We say that a group splits over a virtually nilpotent group i can be decomposed as a nontrivial amalgamated product over a virtually nilpotent group or as an HNN-extension over a virtually nilpotent group Theorem (Rips). Let be a nitely presented group which is not virtually nilpotent and does not split over virtually nilpotent. Then F D(; G)=G is compact (notice that the empty set is compact) Proposition. Let M be a closed (possibly nonorientable) K(?; 1) topological n-maniold with n 3 such that? is word-hyperbolic. Then? is not virtually nilpotent and does not split over virtually nilpotent. Proo. Beore starting the proo we recall several useul algebraic acts. First, note that? is torsion ree since it is the undamental group o an aspherical maniold. Second, any torsion ree virtually cyclic group is, in act, innite cyclic because it has two ends [St, 4.A.6.2, 5.B.1]. Thus, any virtually cyclic subgroup o? is innite cyclic.

12 12 IGOR BELEGRADEK Third, show that any virtually nilpotent subgroup o a torsion ree word-hyperbolic group? is innite cyclic. By the remark above, it suces to prove that any nilpotent subgroup N? is cyclic. Recall that in a word hyperbolic group? the centralizer C x (?) o a nontorsion element x is a nite extension o an innite cyclic group generated by x [CDP, ]. We have < x > C x (N) C x (?) or any x 2 N. Thereore, < x > has nite index in C x (N); so C x (N) is virtually cyclic, hence C x (N) = Z. Being a nilpotent group N must have a nontrivial center Z(N); clearly Z(N) C x (N), so Z(N) = Z. This impies that, or any x 2 N, there exist an integer m such that x m 2 Z(N). Thereore, or any x; y 2 N there is an integer m with the property x m y = yx m, or x m = yx m y?1 = (yxy?1 ) m. Any element o a torsion ree nilpotent group has at most one m-th root, hence x = yxy?1. Thus N must be abelian, that is N = Z(N) = Z as desired. In particular, the group? itsel is not virtually nilpotent because cd? = dim(m) 3 while Z has cohomological dimension one. We now start the proo. Argue by contradiction. Thus, either? =? 1 C? 2 or? =? 0 C where C is innite cyclic and? k is a proper subgroup o?, k 2 0; 1; 2g. First, we assume that either? =? 1 C? 2 and j? :? k j = 1 or k 2 1; 2g or? =? 0 C. (notice that by the denition o HNN-extension? 0 has to have an innite index in?). Then it ollows rom Mayer-Vietoris sequence [Bro VII.9.1] that cd? max(cd? 1 ; cd? 2 ; cdc + 1) (c. [Bro, VIII.2.2.4(c)]). Since j? :? k j = 1,? k is a undamental group o a noncompact n-maniold that is a cover o M, thereore, cd? k < cd? [Bro, VIII.8.1(a)]. As cdc = 1, we get cd? 2; on the other hand cd? = dim(m) = n 3, a contradiction. Second, suppose that? =? 1 C? 2 and or some k, j? :? k j < 1. Say? 1 has nite index in?. This implies j? 2 : Cj < 1 because C =? 1 \? 2. Then? 2 is virtually cyclic and, hence, innite cyclic. We now use the Mayer-Vietoris sequence [Bro VII.9.1] or homology with twisted coecients where n = dim(m)????! H n C????! H n? 1 H n? 2????! H n?????! H n?1 C????! : Since n 3 we get 0 = H n C = H n?1 C = H n? 2. In particular, the inclusion? 1,!? induces an isomorphism in homology. We next show that the map H n (? 1 )! H n (?) induced by the inclusion? 1,!? is the multiplication by j? :? 1 j. Thus j? :? 1 j = 1 which contradicts the assumption that? 1 is a proper subgroup o?. Indeed, being a compact maniold M is homotopy equivalent to a nite CW -complex Y [We]. The universal cover Y ~ o Y is a ree?-space (and, hence, Y ~ is a ree?1 -space). The identity map o Y ~ is equivariant with respect to the inclusion?1,!?, thereore it induces an equivariant map o cell complexes C ( Y ~ )! C ( Y ~ ). Passing to invariants we have a map C ( Y ~ )?1! C ( Y ~ )?. We know that C ( Y ~ )?1 = C ( Y ~ =?1 ) and C ( Y ~ )? = C ( Y ~ =?) = C (Y ). Moreover, the corresponding map C ( Y ~ =?1 )! C (Y ) is induced by a nite cover Y ~ =?1! Y = Y ~ =?. Thus the induced homomorphism in homology with twisted coecients H n ( Y ~ =?1 )! H n (Y ) is the multiplication by j? :? 1 j = deg( ~ Y =?1! Y ) (we recall that Y and ~ Y =?1 are homotopy equivalent to closed n-maniolds, thereore, H n (Y ) = Z = H n ( ~ Y =?1 )). x6. The Main theorem. Let Y be a nite CW-complex and y 2 Y. Let X be a contractible homogeneous Riemannian maniold and G be a transitive group o orientation-preserving isometries. We equip Hom( 1 (Y; y); G) with algebraic (i.e. pointwise convergence) topology. Let be an invariant o continuous maps. We also use the letter or the corresponding invariant o discrete representations Theorem. Let n 2 Hom( 1 (Y; y); G) be a sequence o representations such that the groups n ( 1 (Y; y)) are discrete and torsion ree. Assume that n converges to algebraically and the groups n ( 1 (Y; y)) converge geometrically to a discrete group? geo. Let : Y! X=? geo be a continuous

13 HYPERBOLIC MANIFOLDS AND VECTOR BUNDLES 13 map that induces the homomorphism : 1 (Y; y)! ( 1 (Y; y))? geo. Then there exists a positive integer N such that (1) or every n > N, ( n ) = (), (2) moreover, i either is homotopic to an embedding or ( 1 (Y; y)) =? geo, then ( n ) = () or n > N. Proo. According to 4:11, the group? geo is discrete. Being a limit o torsion ree groups the discrete group? geo is itsel torsion ree by 4:6. Thus? geo acts reely and properly discontinuously on X, so X=? geo is a maniold. We consider the universal covering q : Y ~! Y ; clearly Y ~ has a natural cell structure so that the covering group acts by permuting cells. We x an isomorphism :! (Y; y) by choosing a point ~y 2 q?1 (y). Since Y is a nite complex, one can choose a nite connected subcomplex D Y ~ with q(d) = Y. (Pick a representing cell in every orbit, the union o the cells is a nite subcomplex that is mapped onto Y by q. Adding nitely many cells, one can assume the complex is connected.) The representation denes a continuous map : Y! X=() which is dened up to homotopy by 3.2. The map can be lited to a -equivariant map ~ : Y ~! X o universal coverings. The identity map id : X! X is equivariant with respect to the inclusion (),!? geo, thereore, ~ is equivariant with respect to the homomorphism :!? geo. We denote the composition o and the covering X=()! X=? geo = N induced by the inclusion ()? geo. Let U X be an open relatively compact neighborhood o (D). ~ We are in position to apply the Theorem 4.9. Thus, i n is large enough, there is a sequence o embeddings ~' n : U! V X that is converging to the inclusion and is r n -equivariant. So the map ~' n ~ : D! V is r n -equivariant. By the very denition o r n we have r n = n whenever let hand side is dened. We now extend the map ~' n ~ : D! V by equivariance to a n -equivariant map ~ n : Y ~! X. The map n ~ descendes to n : Y! V= n () X= n (). Notice that by construction n = ' n. Since ' n converge to the inclusion, ' n is orientation-preserving or large n. Thus ( n ) = () or large n (see 2.9). Notice that ( n ) = ( n ). Denote ~w = ~ n (~y) and w = p n ( ~w) where p n is the covering p n : X! X= n ( 1 (Y; y)) = N n and consider the corresponding isomorphism n : n ( 1 (Y; y))! 1 (N n ; w). We need to veriy that the homomorphism n : 1 (Y; y)! 1 (N n ; w) coincide with n n. Start with a loop [!] 2 1 (Y; y); it lits to a path ~! with endpoints ~y and?1 ([!])(~y). Since ~ n is n -equivariant, ~ n (~!) has endpoints ~w and ~ n (?1 ([!])(~y)) = n (?1 ([!])( (~y)) ~ = n ([!])( ~w). Thus n ([!]) = [ n (!)] = n n ([!]), hence, ( n ) = ( n ) as desired. Thereore, or large n, ( n ) = () which proves (1). We are to prove (2) now. Clearly, i ( 1 (Y; y)) =? geo, then =, hence, () = ( ) = () = ( n ) or large n. In case is homotopic to an embedding, so is, hence ( ) = () (see 2.10). Thus () = ( ) = () = ( n ) or large n, and we are done. 6.2 Remark. I Y is a closed smooth maniold and 2dim(Y ) < dim(x), the Whitney embedding theorem implies that any map o Y into X=? geo is homotopic to a smooth embedding. Thus in this case ( n ) = () or large n From now on we assume that G = Isom + (H n F ), the group o all orientation-preserving isometries o H n F. Let F D(; G) Hom(; G) be a subspace o aithul discrete representations. As beore Y is a nite connected CW -compex. A group? is said to satisy the condition (i) i the space F D(?; G)=G is compact; (ii) i? is a nontrivial group without nontrivial normal nilpotent subgroups. For example,? satises (i)-(ii) i? is word-hyperbolic and there is a closed oriented K(?; 1) maniold o dimension n 3. As we explained in 4.12, or nitely generated discrete subgroups o G, (ii) is equivalent to being non-virtually nilpotent. 6.4 Corollary. Let = 1 (Y; y) and let n 2 Hom(; G) be a sequence o representations such that the groups n () are discrete and torsion ree. Let n converge to algebraically. Assume () satises (ii). Then the set ( n )g is nite.

14 14 IGOR BELEGRADEK Proo. This ollows rom 6.1 and Denition. Let : X! Y be a map rom a topological space X to a set Y. We say that is a locally nite-range map i every point x 2 X has a neighborhood U x such that (U x ) is a nite set Corollary. Assume = 1 (Y; y) is a torsion ree group that satises (ii). Then the map : F D(; G)! X B is a locally nite-range map. Moreover, i Y is a closed smooth maniold and 2dim(Y ) < dim(x), then the map : F D(; G)! X B is locally constant. Proo. This ollows rom 6.4 and Corollary. Assume = 1 (Y; y) is a torsion ree group that satises (i)-(ii). Then the image (F D(; G)) is a nite subset o X B. Proo. Since the G-action on H n F is orientation-preserving, (g g?1 ) = () (see 3:3). Thereore, we get a locally nite-range map : F D(; G)=G! X B : (Indeed, take [] 2 F D(; G)=G. Since is S a locally nite-range map we can nd U with (U ) nite. The -image o the open set GU = g2g gu g?1 is a nite set. By the denition o actortopology the image o GU in F D(; G)=G is an open set. This is a neighborhood o [] with a nite image.) Clearly, the compactness o F D(; G)=G implies that (and hence ) is a nite-range map Theorem. Assume = 1 (Y; y) is a torsion ree group that satises (i)-(ii). Let n be a positive integer and let 2 H m (Y ) and 2 H n?m (Y ). Then there is a positive integer K such that (1) or any continuous map : Y! N o Y into an oriented F-hyperbolic n-maniold N that induces an isomorphism o undamental groups, and (2) or any embedding : Y! N o Y into an oriented F-hyperbolic n-maniold N that induces a monomorphism o undamental groups, jh ; ij K. Proo. We apply the corollary 6.7 to the case = " : F D(; G)=G! Hom(H (Y ); H (Y )) to conclude that Im(") is nite. Since jh ; ij = j(?1) n(n?m) h"(); ij we are done Corrolary. Let M be a closed oriented maniold o dimension n 2 such that 1 (M) satises (i)-(ii) and has no torsion (e.g. M is an oriented aspherical maniold and 1 (M) is word-hyperbolic). Then there is a nite subset F H k (M) that contains the Euler class o any topological embedding o M into a complete oriented F-hyperbolic (n+k)-maniold i the embedding induces a monomorphism o the undamental groups. Thus, F contains the Euler class o any oriented vector bundle R k! E! M i E admits a complete F-hyperbolic structure. In particular, only nitely many oriented R 2 -bundles over M admit a complete F-hyperbolic structure. Proo. Let : M! N be an embedding that induces a monomorphism o the undamental groups. Consider a covering p : N ~! N with p 1 ( N) ~ = 1 (M) and lit to an embedding ~ : M! N. ~ According to 2.11 we get e() = e( ) ~ = "()[M] and the result ollows rom the corrolary Remark. Observe that the theorem 0:1 (and, hence 0:2 and 0:4) stated or complete hyperbolic maniolds is (trivially) true or any hyperbolic maniold with injective developing map and torsion ree undamental group. Indeed, recall that a developing map o a hyperbolic maniold N is a local isometry rom the universal cover o N onto an open subset o a rank one symmetric space X that is equivariant with respect to the holonomy representation 1 (N)!? where? is the holonomy group. I a developing map is injective, it is necessarily a dieomorphism onto an open subset o X. Then by equivariance the holonomy representation o N is an isomorphism and the holonomy group? acts reely and properly discontinuously on the image o the developing map. Hence? is a discrete group, and N is