Annals of Mathematics, 147 (1998), 325-355

The Novikov conjecture for groups with finite asymptotic dimension

By GUOLIANG YU*

Dedicated to Professor Ronald G. Douglas on the occasion of his sixtieth birthday

1. Introduction

In this paper we shall prove the coarse Baum-Connes conjecture for proper metric spaces with finite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures.

THEOREM 1. 1. Let F be a finitely generated group whose classifying space BF has the homotopy type of a finite CW-complex. If F has finite asymptotic dimension as a metric space with a word-length metric, then the Novikov con- jecture holds for F.

Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology (page 28, [14]). More precisely, the asymp- totic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = {Ui}iEI of the metric space for which the r-multiplicity of C is at most n + 1; i.e., no ball of radius r in the metric space intersects more than n + 1 members of C [14]. The class of finitely generated discrete groups with finite asymptotic dimension is hereditary in the sense that if a finitely generated group has finite asymptotic dimension as metric space with a word-length metric, then its finitely gener- ated subgroups also have finite asymptotic dimension as metric spaces with word-length metrics (cf. Section 6). This, together with a result of Gromov in [14], implies that finitely generated subgroups of Gromov's hyperbolic groups have finite asymptotic dimension. Currently no example of a finitely generated group with infinite asymptotic dimension and finite classifying space is known. It should also be noted that two different definitions of asymptotic dimension

*Research supported in part by the National Science Foundation.

326 GUOLIANG YU

are introduced by Gromov and it is not known if the two definitions are equal (see page 28-32 of [14] for more details).

The validity of the Novikov conjecture has been established, by a variety of techniques, for many groups [3], [5], [7], [8], [9], [10], [23], [24], [27], most notably for fundamental groups of complete manifolds with nonpositive sec- tional curvature, closed discrete subgroups of Lie groups with finite connected components, and Gromov's hyperbolic groups. Our approach to the Novikov conjecture is coarse geometric in spirit and is based on the descent principle that the coarse Baum-Connes conjecture for a finitely generated group F (as a metric space with a word length metric) implies the strong Novikov conjecture for F if the classifying space BF has the homotopy type of a finite CW-complex [33] (as pointed out by the referee, this descent principle is a C*-algebra ver- sion of various descent principles previously known to topologists [4], [5], [11]). Our main new tool is controlled operator K-theory. Controlled operator K- theory is a refinement of ordinary K-theory, incorporating additional norm and propagation control.

Our result on the coarse Baum-Connes conjecture also implies the Gromov- Lawson-Rosenberg conjecture on the nonexistence of Riemannian metrics with positive scalar curvature for compact K(ir, 1)-manifolds when the fundamen- tal group 7r has finite asymptotic dimension as a metric space with a word- length metric and Gromov's zero-in-the-spectrum conjecture for uniformly con- tractible Riemannian manifolds with finite asymptotic dimension. Recall that Gromov's zero-in-the-spectrum conjecture says that the spectrum of the Lapla- cian operator acting on the space of L2-forms of a uniformly contractible Rie- mannian manifold with bounded geometry contains zero.

We shall also construct a proper metric space with infinite asymptotic dimension for which the coarse Baum-Connes conjecture fails. This indicates that our result on the coarse Baum-Connes conjecture is best possible in some sense.

I thank Xinhui Jiang for many stimulating conversations and John Roe for explaining to me the descent principle. I also thank the referee for helpful comments.

2. The descent principle

In this section we shall briefly discuss the coarse Baum-Connes conjec- ture and the descent principle. We remark that C*-algebras in this paper are complex C*-algebras and K-theory is 2-periodic complex topological K-theory.

Let X be a proper metric space. Recall that a metric space is called proper if every closed ball in the metric space is compact. An X-module is a separable Hilbert space equipped with a *-representation of Co (X), the algebra

NOVIKOV CONJECTURE 327

of all complex-valued continuous functions on X which vaxlish at infinity. An X-module is called nondegenerate if the *-representation of Co(X) is nonde- generate. An X-module is said to be standard if no nonzero function in Co(X) acts as a compact operator.

Definition 2.1. Let Hx and Hy be X and Y-modules, respectively. The su,pport of a bounded operator from HXtoByis defined to be the complement (in X x Y) of the set of all points (x, y) E X x Y for which there exist furlctions f E Co(X),g E Co(Y) suchthat gTf =0, and f(x) 7&0, g(y) 7&0.

Definition 2.2. Let Hx be a X-module; let T be a bounded linear oper- ator acting on Hx. (1) The propagation of T is defined to be: sup{d(x, y): (x, y) E Supp(T)}; (2) T is said to be locally compact if fT and Tf are compact for all f E Co(X).

Definition 2.3 ([31]). Let Hx be a standard nondegenerate Z-module. C*(X) is defined to be the C*-algebra generated by all locally compact oper- ators acting on Hx with finite propagations.

C*(X) does not depend on the choice of the standard nondegenerate X-module HXUP to unnatural isomorphism and its K-theory groups are in- dependent of the choice of HXUP to natural isomorphism [31].

A Borel map f from a proper metric space X to another metric space Y is called coarse if (1) f is proper, i.e., the irlverse image of any bounded set is bounded; (2) for every R > O, there exists R' > O such that d(f(x), f(y)) < R' for all x, y E X satisfying d(x, y) < R.

LEMMA 2.4. Let f be as above; let Hx and Hy be respectively standard nondegenerate X and Y-mod?;les. For any E > O, there exists an isometry Vf from Hx to Hy such that

SUpp(Vf) 5 {(x,y) E X x Y: d(f(x),y) < £}

Proof. The *-representation of Co(Y) on Hy can be extended to a *- representation of the algebra of all bounded Borel furletions on Y. There exists a Borel correr {Yi} of Y such that (1) YqnYj = 0 if i 7& j; (2) diameter(Y) < £/2 for all i; (3) each Yq has nonempty interior.

(3), together with the standardness of Hy, implies that %ytH is infinite dimensional, where XYE is the characteristic function of Y. This implies that there exists an isometry Vf from Hx to Hy such that Vf maps %f_1(E)Hx to xytHy, where Xf-l(E) is the characteristic function of f-l(Y) and the *- representation of Co(X) on Hx is extended to a *-representation of the algebra of all bounded Borel functions on X. It is not difficult to see that Vf satisfies the required conditions. O

328 GUOLIANG YU

Vf gives rise to a homomorphism ad(Vf) from C* (X)+ to C* (Y)+ defined by:

ad(Vf)(a + ci) = Vf aV7 + cI

for all a E C*(X) and c E C, where C*(X)+ and C*(Y)+ are obtained from C*(X) and C*(Y) by adjoining identities. Also ad(Vf) induces a homomor- phism ad(Vf)* from Ki(C*(X)) to Ki(C*(Y)).

The following lemma says that ad(Vf)* does not depend on the choice of Vf.

LEMMA 2.5 ([19]). Let f, Hx and Hy be as in Lemma 2.4. If V1 and V2 are two isometries from Hx to Hy such that for some E > 0

Supp(Vi) C {(x,y) E X x Y: d(f (x),y) &lt; E}

for i = 1,2, then

ad(Vi)* = ad(V2)*: Ki (C* (X)) -+Ki (C* (Y))i

See Section 4 of [19] for a proof. Recall that the K-homology groups Ki(X) = KKi(Co(X), C) (i = 0,1)

are generated by certain cycles modulo a certain equivalence relation [23]: (1) A cycle for Ko(X) is a pair (Hx, F), where Hx is an X-module and

F is a bounded linear operator acting on Hx such that F*F - I and FF* - I are locally compact, and OF - Fq is compact for all q E 0o(X);

(2) A cycle for K1 (X) is a pair (HX, F), where Hx is an X-module and F is a self-adjoint operator acting on Hx such that F2 _ I is locally compact, and OF - Fq is compact for all q E Co (X).

In the above description of cycles for Ki(X), Hx can always be chosen to be standard and nondegenerate. See [18], [20] for more details of the above description of K-homology.

Next we shall define the index map from Ki (X) to Ki (C* (X)). Let (Hx, F) be a cycle for Ko(X) such that Hx is a standard nondegenerate X-module. Let {Ui}i be a locally finite and uniformly bounded open cover of X and {4i}i be a continuous partition of unity subordinate to the open cover. Define

F1~ 1?FO F' = E¢i ~

where the infinite sum converges in strong topology.

LEMMA 2.6. Let F and F' be as above.

(1) F' has finite propagation;

(2) jjF'JJ &lt; 41IF11.

329 NOVIKOV CONJECTURE

Proof. (1) is obvious. To prove (2), we write F = ,4 -1 CmTm such that Cm is a complex number satisfying tc2 l < 1 for each m, and Tm is a nonnegative operator acting on Hx satisfying tlTmll < tlFtt for each m. It is enough to prove that

n

_ _

' ¢>i2Tm¢>i2 < tlT i_1

1 1

for all n and m. But this follows from the fact that .n 1 Xi2TmXi2 is nonneg- ative and the following estimation:

( ( 0z2 Tm0z2 ) h) h) = S(Qz2 Tm+z h) h) = (Tm+ h Qz h) i-1 i=l i=l

n 1 1 n < E IlTm 11 (+i2 hv Xi2 h)-E IlTm 1l (+ihv h) = IlTm 11 (hv h)

i=l i=l

for all h E Hx.

It is easy to verify that (Hx, F') is equivalent to (Hx, F) in Ko(X). By the above lemma it is not difficult to see that F' is a multiplier of C*(X) and F' is a unitary modulo C*(X). Hence F' gives rise to an element [F'] in Ko(C*(X)). We define the index of (HX,F) to be [F]. Similarly we can define the index map from K1(X) to K1(C*(X)).

Let C be a locally finite and uniformly bounded cover for X. The- nerve space NC associated to C is defined to be the simplicial complex whose set of vertices equals C and where a finite subset {Uo,...,Un} C C spans an n-simplex in NC if and only if n=OUi 78 0. Endow NC with the spherical met- ric. Recall that the spherical metric on the simplicial complex NC is defined as follows. Identify every simplex {Uo, . . ., Un} in Nc with S+ = {(xo, , xn) E

Rn+l xi > 07 n=O xi2 = 1} by:

n / n X 1/2 / n X 1/2\

E ttUi t t0/ tE ti2) ) . . . ) tn/ tE t2) | )

i=O K i=O t=0 /

where ti > O, q=oti = 1. The standard spherical metric on the simplex {Uo, . . ., Un} is defined to be the metric induced from the standard Riemannian metric on the nth unit sphere sn. The spherical metric on NC is defined to be the maximal metric whose restriction to each simplex is the standard spherical metric. If x and y lie in different connected components of Nc, then we define d(x, y)-oo. Use of the spherical metric is necessary to avoid certain pathological phenomena when n goes to infinity.

A sequence of locally finite and uniformly bounded covers {Ck}k=l of X is called an anti-Cech system of X [31] if there exists a sequence of positive

330 GUOLIANG YU

numbers Rk -4 xc such that for each k,

(1) every set U E Ck has diameter less than or equal to Rk; (2) any set of diameter Rk in X is contained in some member of Ck+1.

An anti-Cech system always exists [31]. By the property of the anti-Cech system, for every pair k2 > k1, there

exists a simplicial map ikik2 from Nckl to NCk2 such that ikik2 maps a simplex {Uo,...,Ul} in

Nck, to a simplex {U6, . . ., UI} in NCk2 satisfying Ui C U' for all 0 < i < n. Also, ikik2 gives rise to the following inductive systems of groups:

(ik1k2)*: K(Nck1 ) - K(Nck);

ad(Viklk2)*: Ki(C*(NCk,)) + Ki(C*(Nck))

The following conjecture is called the coarse Baum-Connes conjecture,

CONJECTURE 2.7 ([31]). The index map induces an isomorphism from liMkjo. Ki(Nck) to K(C (X)) = limk, Ki(C* (Nck))

It is not difficult to see that the coarse Baum-Connes conjecture for X does not depend on the choice of the anti-Cech system.

The following descent principle [33] is a C*-algebra version of various descent principles previously known to topologists [4], [5], [11].

THEOREM 2.8 ([33]). Let F be a finitely generated group. If the classifying space Br of r has the homotopy type of a finite CW-complex, then the coarse Baum-Connes conjecture for F (as a metric space with a word-length metric) implies the strong Novikov conjecture for F.

Recall that the word-length metric on a finitely generated group r is defined as follows. By choosing a finite generating set S for r, for any -y E r, we can define its length Is(-y) to be the smallest integer n such that there exists {si}L 1 for which -y = S... Sn and either si or sj 1 S for all i. The word-length metric ds on r is defined by:

ds(-yl,-y2) = Is(-y--y2)

for all -yi, % E r. It is not difficult to see that, for any two finite generating sets S1, S2 of F, (r, dsl) is quasi-isometric to (r, dS2).

3. Localization and the obstruction group

In this section we shall recall the localization technique introduced in [39] and introduce an obstruction group to the coarse Baum-Connes conjecture.

331 NOVIKOV CONJECTURE

Definition 3.1 ([39]). (1) Let X be a proper metric space. The localization algebra CL(X) is defined to be the C*-algebra generated by all bounded and uniformly norm-continuous functions f from [O, oo) to C*(X) such that

propagation(g(t)) O as t oo;

(2) Letting f be an element in CL(X), we define its propagation to be

suptElooo)propagation(f (t))

Next we shall define a local index map from the K-homology group Kz (X) to the K-theory group Ki(CL(X)). For each positive integer n, there exists a locally finite open cover {Un,i}i for X such that diameter(Un,i) < l/n for all i. Let {(g)n,i}i be a continuous partition of unity subordinate to {Un,i}i. Let (Hx, F) be a cycle for Ko(X) such that Hx is a standard nondegenerate X-module. Define a family of operators F(t) (t - [O, oo)) acting on Hx by:

1 1 1 1

F(t) = ,((n-t)+2,iFf2,i + (t n + 1)¢>2+1,iF¢>n+1,i) - i

for all positive integers n and t E [n-1,n], where the infinite sum converges in strong topology. Lemma 2.6 implies that F(t) is a bounded and uniformly norm-continuous function from [O, oo) to the C*-algebra of all bounded oper- ators acting on Hx. Notice that

propagation(F(t)) > O as t oo.

Using the above facts it is not difficult to see that F(t) is a multiplier of CL(X)

and F(t) is a unitary modulo CL(X). Hence F(t) gives rise to an element [F(t)] in KO(CL(X)) We define the local index of the cycle (Hx, F) to be [F(t)]. Similarly we can define the local index map from K.(X) to KI(CL(X)).

Let e be the evaluation homomorphism from CL(X) to C*(X) defined by:

e(f) = f(°) for every f E CL(X)

We have the following commuting diagram:

K* (X)

IndL / \Ind

*

K*(CL(X)) > K*(C*(X)))

where Ind and IndL are respectively the index and local index map. We define the dimension of a simplicial complex X to be the smallest

integer n such that the dimension of each simplex of X is at most n.

332 - GUOLIANG YU

THEOREM 3.2 ([391). Let X be a simplicial complex endowed with the spherical metric. If X is of finite dimension, then the local index map from Ki (X) to Ki (CL (X)) is an isomorphism.

This theorem can be proved by induction on the i-skeleton X(i) (as a metric subspace of X) and a Mayer-Vietoris sequence argument (cf. [391).

(We would like to correct a misleading notational error in the proof of the above theorem in [391 (ProofofProposition 3.7, page 314, [391). Let F(x,t), ti?j and £i be as in the proof of Proposition 3.7 in [391. Let Vk,i be an isometry from HX to Hx such that Supp(Vk,i) C {(x, y) E X x X: d(F(x, tk,i), y) < £i},

and Vk,i = I if F(x,tk,i) = x for all x E X. Define a family of isometries Vk(t) (t E [O,oo)) from HX @ HX to HX @ HX by: Vk(t) = R(t-i + 1) (%kxi-l @ Vk,i)R*(t-i + 1) for all i-1 < t < i and i > 1, where

R( ) t cos(rt/2) sin(Tt/2)8 t-sin(7rt/2) CoS(7Tt/2)J

Finally, for any unitary u representing an element in K1(CL(X)), we define a = k>O Ad(Vk)(u)(u l @ I), b = k>O Ad(%k+l)(u)(u 1 @ I), C = I k>l Ad (Vk ) (u) (u- l @ I), where Ad (Vk ) (g + CI) = Vk (t) (g (t) @ O) Vk* (t) + CI for any g E CL(X) and c E (C. With the above correction, the main argument in the proof of Proposition 3.7 in [391 remains the same.)

Let f be a proper map from a proper metric space X to another proper metric space Y. Assume that f is uniformly continuous, i.e. for any £ > O, there exists O$ > O such that d(f(x), f(y)) < £ for x, y E X satisfying d(x, y) < 6. Let {£k}k be a sequence of positive number such that £k > O as k > w. By Lemma 2.4, for each k > O, there exists an isometry Vk from a standard nondegenerate X-module HX to a standard nondegenerate Y-module Hy satisfying

SUPP(Vk) C {(X,Y) E X X Y: d(f(x),y) < £k}@

Define a family of isometries Vf (t) (t E [O, oo)) from Hx HX to Hy Hy 1

by:

Vf(t) =R(t-k+1)(Vk@Vk+l)R*(t-k+1)

for all k-l < t < k, where

R(t)= ( _sin(frt/2) cos(frt/2) )

Vf (t) induces a homomorphism Ad(Vf ) from CL (X)+ to CL (Y)+ Q3 M2 ((% defined as follows:

Ad(Vf ) (u + CI) = Vf (t) (u(t) ffl O)Vf* (t) + CI

for any u E CL(X) and c E (S, where CL(X)+ and CL(Y)+ are obtained from CL(X) and CL(X) by adjoining identities. Notice that Ad(Vf)(u+cI) is uniformly norm-continuous in t although Vf (t) is not norm-continuous.

NOVIKOV CONJECTURE 333

We have the following useful lemma:

LEMMA 3.3. Let X, Y, f, {Ek}k and Ad(Vf) be as above. Ad(Vf) is a homomorphism from CL(X)+ to CL(Y) + 0 M2 (C) satisfying

propagation(Ad(Vf ) (u)) < propagation(u) + 2 SUPk (6k)

Definition 3.4. Let X be a proper metric space. CLo(X) is defined to be the C*-algebra generated by all bounded and uniformly norm-continuous functions f from [0, oo) to C*(X) such that propagation(f(t)) -> 0 as t > 0o,

f(0) = 0.

We have the following short exact sequence:

O >- CL ,O(X)

- > CL*(X) >- C*(X) O- .

This and Theorem 3.2, imply the following:

THEOREM 3.5. Let X be a proper metric space. Assume that there exists an anti-Cech system {Ck}k for X such that Nck is finite dimensional for all k. Then the coarse Baum-Connes conjecture holds for X if and only if

lim Ki(CLO(NCk)) = 0 k--*oo

for i= 0,1.

In the above theorem, the inductive system of groups is given by:

Ad (1k1k2)* Ki (CLo(Nckl)) ) Ki(CLO(Nck2))v

where Zk1k2 is as in Conjecture 2.7, and Ad(1iklk2) is as in Lemma 3.3. Recall that. X is said to have (locally) bounded geometry if there exists a

subspace r such that (1) there exists c > 0 for which d(x, r) < c for all x E X; (2) for any r > 0, there exists a natural number n(r) such that the number of elements in Bp('y,r) = {1y' E r: d (y',gy) < r} is at most nr(r) for every y E r. If X has (locally) bounded geometry, then there exists an anti-Cech system {Ck}k for X such that Nok is finite dimensional for all k.

For obvious reason limkoo Ki(Q W o(Nck)) is called the obstruction group to the coarse Baum-Connes conjecture.

4. Controlled obstructions: QP6,rsk(X), QU6,r,s,k(X)

In this section we shall introduce and study QP6S,r,s,k (X) and QUS,r,s,k (X),

which can be respectively considered as controlled versions of Ko(CLZ o(X) 0

Co ((O. i)k)) and K1 (CL' 0(X) 0 Co ((0, i)k)).

334 GUOLIANG YU

Let A be a C*-alg;ebra and $ be a positive number. An element p in A is

called a $-quasi-projection if

p* = p llp2 | Pll < 6

Similarly an element u in A is called a b-quasi-unitary if

llu*u-zII < a, Iluu*-III < a.

Let X be a proper metric space; let CL O(X)+ be the C*-algebra obtained

from CL O(X) by adjoining; an identity. Let O$, r and s be positive numbers, k

and n be nonnegative integers. Define QE5XrXsXk(cl o(X)+ (3 Mn((C)) to be the

set of all continuous functions from [O, llk to CL O(X)+ C) Mn((C) such that:

(1) f(t) is a $-quasi-projection and propagation (U(t)) < r for all t E

[O, Ilk;

(2) 1 is piecewise smooth in ti and 118fi (t)ll < s for all t = (tl,...,tk) E

[O, llk;

(3) || f (t)-Pm || < 6$ for all t E bd([O, l]k), the boundary of [O, l]k in Rk

where Pm = I 9 * * ffl I ffl O ffl * * ffl O with m identities; (4) x(f(t)) = Pm) where gr is the canonical homomorphism from CL O(X)+

Mn ((S) to Mn (C) @

Definition 4.1. QP^rXsXk (X) is defined to be the direct limit of

QP6,r,s,k (CL,O (X) + (S) Mn ((C) ) under the embedding: p > p ffl 0.

Similarly we define QUyXrXsXk(CL O(X)+ (3 Mn((C)) to be the set of all con-

tinuous functions from [O, l]k to CL O(X)+ (8) Mn((C) such that:

(1) f(t) is a $-quasi-unitary and propagation(U(t)) < r for all t E [0, l]k;

(2) k is piecewise smooth in ti and 11 afi (t) 11 < s for all t E [0) l]k;

(3) llf(t)-Ill < 6 for all t E bd([O) l]k); (4) lr(k(t)) = I, where 7r is the canonical homomorphism from CL O(X)+Q3

Mn (¢:) to Mn ((C) @

Definition 4.2. QUaXrXSXk (X) is defined to be the direct limit of

QU6,r,s,k(CL,o(X)+ (3 Mn((C)) under the embeddirlg: u u (13 I.

Definition 4.3. Let p and q be two elements in QPEXrXSXk(CLO(X)+ (8)

Mn((C)) Now p is said to be (E, r, s)-equivalent to q if there exists a piecewise

smooth homotopy a(t') (t' E [O, 1]) in QP6XrXsXk(CL o (X)+ Q3 Mn ((C)) such that

(1) a(O) = p and a(l) = q; (2) llat(tt)ll < s.

The above concept can be used to define the concept of (E, r, s)-equivalence between elements in QP^rXSXk (X) .

Notice that if 6 < 1OO, then there exist universal constants co and so such

that (1) any p E QP6,r,s,k (CL O (X)+ 'S Mn(¢:)) is (10E, r, cos + so)-equivalent to some q in QPlO6XrXcos+soXk(cLXo(x)+ (23 Mn(C)) satisfying q(t) = 7r(q(t)) =

NOVIKOV CONJECTURE 335

Pm for some m and all t E bd([0, 1]k); (2) if p and q are two elements in

QP6,rsk(CQ O(X)+ (0 Mn((C)) such that p is (6,r,s)-equivalent to q, p(t) =

ir(p(t)) and q(t) = 7r(q(t)) for all t E bd([O, 1]k), then there exists a homo-

topy a(t') (t' E [0, 1]) in QPlO6,rcos+sok(CL O(X)+ 09 Mn (C)) satisfying a(O) =

p, a(1) = q, and (a(t'))(t) = ir((a(t'))(t)) for all t' e [0, 1] and t E bd([0, i]k).

By the following lemma, QP6,r,s,k(X) can be considered as a controlled version of Ko(CL,0(X) 0 Co((O, 1)k)).

LEMMA 4.4. Let 0 < 6 < 1/100; let f be a continuous function on ]R satisfying f (x) = 1 for all x E [1/2,3/2], and f (x) = 0 for all x E [-1/5,1/5].

(1) For any p E QPS,r,s,k(X), f (p) is a projection and defines an element

[f (p)] in Ko(C,0 (X) 0 Co((0,1)k)); (2) Let p and q be elements in QPSrsk (X) . If p is (6,r,s) -equivalent to q,

then [f (p)] = [f (q)] in Ko(CLO(X) 0 Co((0,1)k));

(3) Every element in KO(CLO(X) 0 C0((O,1)k)) can be represented as

[f (Pi)] - [f (P2)], where P1,P2 E QPS rsk(X) for some r > 0 and s > 0.

Proof. (1) and (2) are straightforward. To prove (3), note that every ele-

ment in Ko(CL 0(X) 0 Co((O, I)k)) can be represented as [qi] - [q2], where q1 and q2 are projections in (CL O(X) 0 Co((0, I)k))+ 0 Mn(C). By an approx- imation argument, there exist P1 and P2 in QP6/l0,rsk(C!O(X)+ 0 Mn(c)) for some r > 0 and s > 0 such that I1ii - qiI1 < 6/10 for i = 1,2. Let

Pi(t') = t'pi + (1 - t')qi for i 1, 2, and t' E [0, 1]. It is not difficult to see

(Pit')= Pi (t/)v I(Pi(t'))2 _Pi (t')l < 6

for i = 1, 2, and t' e [0, 1]. It follows that f (pi(t')) is a homotopy of projections for i = 1, 2 satisfying f (pi(0)) = qi, f (pi(1)) = f (pi). This implies (3). E

Similarly we can define the concept of (6, r, s)-equivalence between ele- ments in QU6,r,s,k (CL O(X) + 09 Mn (C)) and QU6Srsk(X). It is also not difficult

to verify that if 6 < 1 , then there exist universal constants cl and si such that (1) any u E QU6,r,s,k(C2,o(X)+ 0 Mn(C)) is (106, r, cis + sl)-equivalent to some v in QU106,rcis+s1,k(CZO(X)+ (0 Mn(C)) satisfying v(t) = ir(v(t)) = I

for all t E bd([0, 1]k); (2) if u and v are two elements in QU6,rsk(CLO(X)+ 0

Mn(C)) such that u is (6, r, s)-equivalent to v, u(t) = ir(u(t)) and v(t) =

ir(v(t)) for all t E bd([0, 1]k), then there exists a homotopy a(t') (t' E [0, 1])

in QU1O6,rcis+s1,k(CLO(X)+ 0 Mn(C)) satisfying a(0) = u, a(1) = v, and

(a(t'))(t) = 7r((a(t'))(t)) for all t' E [0, 1] and t E bd([0, i]k).

By the above discussions and the following lemma, QU6,r,s,k(X) can be

considered as a controlled version of K1(CL O(X) 0 Co((0, 1)k)).

336 GUOLIANG YU

LEMMA 4.5. Let O < 6 < 1/100. (1) Every U E QUE,r,s,k(X) satisfying u(t) = 7r(u(t)) = I for all t G

bd([O,1]k) defines an element [u] in K1 (CL O(X) (3 CO((O,1)k)); (2) Let u and v be elements in QUaXrXSXk(X) satisfying u(t) = 7r(u(t)) = I

and v(t) = 7r(v(t)) = I for all t E bd([O,l]k). If u is (E,r,s)-eqtbivalent to v, then [u] = [v] in K1 (CL O(X) X CO((O,1)k));

(3) Every element in K1 (CL O (X) X CO ( (O, 1 ) k ) ) can be represented as [u], where U E QU6,r,S,k(X) for some r > O and s > O and u(t) = 7r(u(t)) = I for all t E bd([O,l]k).

The proof of the above lemma is similar to that of Lemma 4.4 and is therefore omitted.

LEMMA 4.6. Let O < 6 < 1/100. If p and q are two (E,r,s)-eqtbivalent ele- ments in QPa,r,s,k (CL O (X)+ fg Mn(¢:)) 7 then there exists u E

QuaXcl(6Xs)rXc2(s)k(cLXo(x)+ X Mn(¢:)) stbeh that IIP-u*qu|| < C3(S)E, where c1(E,s) depends only on 6 and s, c2(s) and C3(5) depend only on s.

Proof. Let a(t) be the homotopy in Definition 4.3. There exists a parti- tion:

°=to <tl < *@@ <tmS = 1

such that lla(ti+l)-a(ti) 11 < 1/100,

where mS depends only on s. Consider

ui = [(2a(t'i+1)-I)(2a(ti)-I) + I]/2. We have

I-ui = (2a(ti+l)-I)(a(ti+l)-a(ti)) + 2(a(ti+l)-a2(t'+l))

This implies that [lI-ui ll < 1/10.

It follows that

(A) III-ui*ui 11 < 3/10.

We also have

(B) lla(ti+l)ui-uia(ti) ' Il(a2(ti+l)-a(ti+l))a(ti)ll + lla(ti+l)(a2(ti)-a(ti))

+ lla2(t'+l)-a(ti+l)ll + lla2(ti)-a(ti)ll < 66. Let Pl(x) be the Ith Taylor polynomial for g=; at 0. Choose lo such that

+/1-x 10 210ms

337 NOVIKOV CONJECTURE

for all x E [0, 3/10]. Let

Wi = uiPtO(I-Ui*Ui)

Define U = Wms-1 * * * wo

Notice that Pl (x) has nonnegative coefficients and the sequence {Pl (x)}l is uniformly bounded on [0, 3/10]. This, together with the chain rule, implies that

u - <C2(5)

for some C2(S) depending only on s. By the choice of lo it is not difficult to verify that u is a b-quasi-unitary. (A), (B) and the self-adjointness of a(t) imply

llui*uia(ti)-a(ti)ui*ui 11 < 206.

Hence

a(ti)(I-Ui*Ui)n-(I-ui*ui)na(ti)ll < Il(a(ti)ui*ui-uiui*a(ti))(I-Ui*Ui)n 111

+ Il(I-ui*ui)(a(ti)ui*ui-uiui*a(ti))(I-Ui*Ui)n 211

* * * + Il(I-ui*ui)n-l(a(ti)ui*ui-uiui*a(ti))

< 20En(3/lO)n-l.

This, together with the definition of wi, implies that

lia(ti+l)wi-wia(ti) 11 < b6)

where b is a universal constant. Now it is not difficult to see that u satisfies the desired properties. E1

For any proper metric space X, let GQP^rXSXk(X) be the set of formal dif- ference p-q, where p,-q E QP^,r,s,k(X), and gr(p) = gr(q). Two elements p-q and p'-ql in GQP6,r,s,k (X) are said to be (E, r, s)-equivalent if p @ ql and p' (13 q are (E, r, s)-equivalent. It is not difficult to see that there exists a universal con- stant so such that any element p-q in GQP^rXSXk(X) is (10E, r, s+so)-equivalent to an element p'-Pm for some p' E QP^,r,s,k(X) and some nonnegative integer m, where Pm = I ffl ffl I @ O ffl ffl 0 with m identities.

An element p-q in GQP^,r,s,k(X) is said to be (E, r, s)-equivalent to 0 if p (13 (I (13 0) is (E, r, s)-equivalent to q (13 (I (13 0).

For any u E QUaXrXSXk(X)) let zt(u) be the homotopy connecting I (13 I to u (13 u* obtained by combining the linear homotopy connecting I (13 I to uu* (13 I with the rotation homotopy connecting uu* (13 I to u (13 u*:

(u @ I)R(t)(U* (13 I)R*(t),

338 GUOLIANG YU

where

R'(t' cos(7rt/2) sin(7rt/ 2) \ - sin(7rt/2) cos(rt/2) 9

Let et(u) = Zt(u)(I @ 0)Zt (u).

We define a map 0 from QU6,r,s,k(X) to GQP1006,lOOr,0oo(s+1),k+1(X) by:

O(u) = et (u) - (I D O),

where t is incorporated as the (k + l)th suspension parameter. The following result can be considered as a controlled version of a classical

result in topological K-theory.

LEMMA 4.7. 0: QUS,r,s,k(X) -* GQPo100,lOOrlOOsk+1(X) is an asymp- totic isomorphism in the following sense:

(1) For any 0 < 6 < 1/100,r > O,s > 0, there exist 0 < 61 < 6,0 < ri < r and si > 0, such that if two elements u and v in QU6j,ri,s,k(X) are (6i,ri,s)- equivalent, then 0(u) and O(v) are (6,r,si)-equivalent, where 61 depends only on 6; ri depends only on r; and s1 depends only on s;

(2) For any 0 < 6 < 1/100,r > O,s > 0, there exist 0 < 62 < 6,0 < r2 < r and S2 > 0 for which if u and v are two elements in QUS2,r2,sk(X) such that O(u) and O(v) are (62,r2,s)-equivalent, then u and v are (6,r,s2)-equivclent, where 62 depends only on 6 and s; r2 depends only on 6,r and s; and S2 depends only on s;

(3) For any 0 < 6 < 1/100,r > O,s > 0, there exist 0 < 63 < 6,0 < r3 < r and S3 > 0, such that for any p Pm E GQPS3,r3,Sk+1(X), there exists u C QUSrS3,k(X) for which 0(u) is (6,r,s3)-equivalent to p - Pm, where 63 depends only on 6 and s; r3 depends only on 6, r and s; and S3 depends only on s.

Proof. (1) Let w(t) be the homotopy realizing the (61, ri, s)-equivalence between u and v. Let a(t) be the homotopy connecting I to v*u obtained by combining the linear homotopy between I and u*u with the homotopy w(t)*u; let b(t) be the homotopy connecting I to vu* obtained by combining the linear homotopy between I and uu* with the homotopy w(t)u*. Define

Xt = Zt (v) (a(t) 0 b(t))zt* (u),

where Zt is as in the definition of the map 0. We have

xo=I, jjx1-III < c61,

lIxtet(u)x*- et (v) II < c6l,

where c is a universal constant. Now (1) follows from the above conditions by choice of appropriate 61, rl and sl.

339 NOVIKOV CONJECTURE

(2) Choose 62 < 6/101°(l + 101°c3(s)),

where C3(S) iS as in Lemma 4.6. By Lemma 4.6 there exists x in QuE2Xcl(62Xs)r2Xc2(s)kil(cLXo(x)+ 3 Mn(¢:)) (for some n) such that

llxet(u ffl I)x*-et(t @ I)|| < c3(s)62.

It follows that

(A) liZt (v @ I)xtZt(u ffl I)(I @ O)-(I e °)Zt (v @ I)xtzt(u @ I)11 < 103c3(s)62, where x is identified with a piecewise smooth family of elements Xt (t E [0,1]) in Q U62 ,cl (62 ,s)r2 ,c2 (S) ,k (CL,O (X) + 8) Mn ((C) ) * Write

(B) Zt (v @ I)xtzt(? @ I) ( ht dt )

(A) implies that

(C) ||gtil < 1000C3(s)62, ||ht|| < 1000C3(s)62

(B), together with the properties of xt and the definitions of zt(v (E3 I) and Zt(1b @ I), implies that

(D) llco ItI < 100U$2, llcl-(v ffl I)*(u e I)ll < lol°E2.

Now, by choice of appropriate r2 and s2, (2) follows from the existence of the homotopy ct satisfying (D).

(3) Choose 63 < 6/101° (1 + lOc3 (s)),

where C3(5) is as in Lemma 4.6. Without loss of generality, we can assume p(O) = p(l) = Pm = I (13 O) where p is identified with a piecewise smooth family of p(t) e QpE3Xr3XsXk(x) (t E [0,1]). By the proof of Lemma 4.6 there exists a homotopy w(t) in QU^3Xcl (^3,s)r3XC2(s)k(x) such that

w(O) = Iv llw(t)(I ffl 0)W(t)*-p(t)ll < C3(S)d6

It follows that

llw(l)(I (E3 O)-(I ffl O)w(l)ll < lOc3(s)U$3. Hence we can write

( h v )

where llgll < lOc3(s)63, llhll < lOc3(s)63. Define

Yt = (w(t) (33 I) (I @ Zt* (v)w* (t)) (Zt* (1b) @ I)

340 GUOLIANG YU

We have

(A) So = I ffl I, IIY1-(I ffl I)ll < 105c3(s)63,

(B) |lYt(et(?s) (13 O)Yt-(p(t) @ °)1l < 105c3(s)63.

Now, by choice of appropriate r3 and S3, (3) follows from the existence of Yt satisfying (A) and (B). O

Let X and Y be two proper metric spaces; let f and g be two proper Lipschitz maps from X to Y. A continuous homotopy F(t,x) (t E [0,1]) between f and g is said to be strongly Lipschitz if:

(1) F(t, x) is a proper map from X to Y for each t; (2) d(F(t, x), F(t, y)) < Cd(x, y) for all x, y E X and t E [O, 1], where C

is a constant (called the Lipschitz constant of F); (3) F is equicontinuous in t, i.e. for any E > O, there exists v$ > O such

that d(F(tl, x), F(t2, x)) < E for all x E X if Itl-t21 < v$; (4) F(O, x) = f (x), F(1, x) = g(x) for all x E X. Note that the above condition of strong Lipschitz homotopy is stronger

than the condition of Lipschitz homotopy introduced by Gromov (see page 25 of [14] or [38]).

Also, X is said to be strongly Lipschitz homotopy equivalent to Y if there exist proper Lipschitz maps f: X ) Y and f1: Y ) X such that f1f and ffi are strongly Lipschitz homotopic to idX and idy, respectively.

LEMMA 4.8. Let f and g be two proper Lipschitz maps from X to Y. Assume that f is strongly Lipschitz homotopic to g. There exists So > O,Co > ° such that for any {lb E QU,r,s,k(X)) there exists a homotopy w(t') (t' E [0,1]) in Quco6XcorXco(s+l)k(cLXo(Y)+ (23 Mn(¢:)) for some n for which

W(0) = Ad(Vf)(?b) (13 I,

w ( 1 ) = Ad(Vg ) (?s) (13 I,

and llwt(tt)ll < So, where Ad(Vf) and Ad(Vg) are as in Lemma 3.3, So and Co depend only on the Lipschitz constant C of the strong Lipschitz homotopy F between f and g. ;

Proof. Choose {ti,j}izo,jzo C [0,1] satisfying (1) to,j = O, ti,j+1 < ti,j, ti+1,i > ti,j; (2) there exists a sequence Nj ) x such that ti,j = 1 for all i > Nj, and

Nj+1 > Nj for all j; (3) d(F(ti+1,i, x), F(ti,j, x)) < ej = r/(j + 1), d(F(ti,j+1, x), F(ti,j, x)) <

ej for all x E X.

341 NOVIKOV CONJECTURE

For example, we can take

t _ f i/(Nj+N) if i < Nj+N, t2 t 1 if i > Nj+N,

where N is some large positive number. Let fi,j(x) = F(ti,j,x) for all x E X. By Lemma 2.4 there exists an

isometry Vfi j from HX to Hy such that

supp(Ffi j) C {(x, y) E X x X: d(fi,j(x), y3 < r/(l + i + j)}.

For each i > O, define a family of isometries Vi(t) (t E [O, x)) from Hx @

HX to Hy 33 Hy by:

Vi(t) = R(t - j)(Vfi j d3 Vfi j--l )R*(t - j)

foralltE [j,j+l],where

( ) V-sin(7rt/2) cos(7Tt/2) cos(7Tt/2) sin(7rt/2) 8

Consider

tso(t) = Ad(Vf)(?s) = Vf(t)(?s(t) 33 I)Vf*(t) + (I-Vf(t)Vf*(t));

tboo(t) = Ad(Vg)(?s) = Vg(t)(?s(t) 33 I)Vg*(t) + (I-Vg(t)Vg*(t)); tbi(t) = Ad(Vi)(?s) = Vi(t)(ls(t) 33 I)Vi*(t) + (I-Vi(t)Vi*(t))

for i > O. Notice that Xbi(t) is uniformly continuous in t although Vi(t) is not continuous in t.

For each i, define ni to be the largest integer j satisfying i > lVj if {j: i > Nj} 7& 0, and define ni to be O otherwise. We can choose Vfij in such a way that tsi(t) = tsoo(t) when t < ni.

Define

( tsi(t)(tsoo(t))* if t > ni;

wi(t) = ?} (ni-t)I + (t ni + l)?si(t)(lsoo(t))* if ni-1 < t < ni;

t I if O < t < ni-1.

Consider

a = ffl2°°o(wi @ I); b = iti°°o(wi+l 33 I); c = (IfflI) @2°°-1 (Wi @ I),

where a, b and c act on the standard and nondegenerate Y-module H°y° = fflz_o((Hy (13 Hy) (@ (Hy 33 Hy)).

By (2), (3) and the construction of a, b and c, we know that a, b and c are elements in QUClE,Clr,Cl(s+l),k(y) for some constant C1 depending only on C.

342 GUOLIANG YU

Let Vivi+l(t') = R(t')(Vi @ Vi+l)R*(t')

for all t' E [0,1]. Define

ui,i+l(t ) = Vi,i+l(t')((u fS I) ffl I)Vi,*i+l(t') + (I-Vi,i+l(t')Vi*i+l(t'))

for t' E [0,1]. Notice that

ui,i+l(O) = (Vi(u fS I)Vi* + (I-ViVi*)) (13 I;

vi i+l(1) = (Vi+l(u fS I)Vi*+l + (I Vi+lVi+l)) (13 I

Using ui,i+l (t/) we can construct a homotopy vl (t') (t' E [0,1]) in QUC2E,C2r,C2s,k(Y) for some C2 > C1 such that

vl(O) _ a,vl(l) = b;

lisl (t') 11 < so

for all t', where so is a universal constant. We can also construct a homotopy v2(t') (t' E [0,1]) in Quc36Xc3rXc3sXk(Y)

for some C3 > Cl such that

V2 (0) = b, v2 (1) = c;

llv2(t/)ll < S1

for all t', where sl is a universal constant. Finally we define w(t') to be the homotopy obtained by combining the

following homotopies: (1) the linear homotopy between

(tsofflI) (132°°-1 (I ffl I) and c*a((vOO 33 I)fflz_l(I 33 I));

(2) V2*(1-tt)a((vOO (13 I)ffl2°°l(I 33 I);

(3) vl (l tt)a((vOO fS I)fflt°° l (I (13 I));

(4) the linear homotopy between

a*a((vOO 33 I)G3z°°1 (I 33 I)) and (tsoo 33 I)i3z_1 (I 33 I).

It is not difficult to see that w(t') is the desired homotopy connecting Ad(Vf ) (?s)0332°°-lI to Ad(Vs) (?s)d3z-lI °

Combining Lemma 4.7 with Lemma 4.6, we have the following result.

LEMMA 4.9. Let X, Y, f and g be as in Lemma 4.8. For any O < 6 < l/lOO,r > O,s > O, there exist O < 61 < 6,0 < rl < r and sl > O such that for

NOVIKOV CONJECTURE 343

any p e QP6b,ri,s,k(X) (k > 1), there exists a homotopy w(t') (t' E [0,1]) in

QP6,rsjk(CLZo(Y)+ 0 Mn(C)) for some n satisfying

w(0) = Ad(Vf)(p 0) E (I E 0),

w(1) = Ad(Vg)(p ED 0) ED (I e o)? and IIw'(t')II < si, where 61 depends only on 6, s, C (C is as in Lemma 4.8); ri depends only on 6, r, s, C; and sl depends only on s and C.

5. Controlled cutting and pasting

In this section we shall prove a controlled cutting and pasting result for

QU6,r,s,k (X).

Definition 5.1. Let X be a proper metric space and Xi (i = 1,2) be metric subspaces. The triple (X; Xi, X2) is said to satisfy the strong excision condition if

(1) X = X1 U X2, Xi is a Borel subset and the interior of Xi is dense in

Xi for i = 1, 2; (2) there exists ro > 0, co > 0 such that (i) for any r < ro, bdr(Xi) n

bdr(X2) = bdr(Xi n X2), where bdr(A) is defined to be {x E X: d(x, A) < r} for any A C X; (ii) for each X' = Xi, X2, X1 n X2, and any r < ro, bdr(X') is strongly Lipschitz homotopy equivalent to X' with co as the Lipschitz constant of the strong Lipschitz homotopies realizing the strong Lipschitz homotopy equivalence.

Let (X; X1, X2) be as in Definition 5.1. We shall first construct a boundary

map a from QUSr,8,k(X) to GQPNo6,N(6)r,Nos,k(X1 nX2), where 0 < 6 < 1/100, N(6) depends only on 6, No is a universal constant, and r < ro/(1 + N(6)) (ro is as in Definition 5.1). Our construction is modeled after the standard con- struction of the boundary map in K-theory (cf. [2] and [26]). The modification is necessary in order to control the propagations.

We shall first make a few remarks about notation. Let Y be a metric subspace of X such that Y is Borel and the interior of Y is dense in Y. Let

HX be as in Definition 2.3. Recall that the *-representation of Co(X) on

HX can be extended to a *-representation of the algebra of all bounded Borel functions. Hence we can define Hy = XyHx, where XY is the characteristic function of Y. Define a Y-module structure on Hy by: fh = f'h for any f E Co(Y), h E Hy C Hx, where f' is any bounded Borel function on X whose restriction to Y is equal to f . Note that f 'h is independent of the choice of f'. Since the interior of Y is dense in Y, Hy is a standard nondegenerate Y-module. For technical convenience we shall choose Hy as the standard

344 GUOLIANG YU

nondegenerate Y-module in the definitions of C*(Y) and CL O(Y) throughout

this section. Each operator T acting on HY defines an operator (still denoted

byT throughout this section) on HX by: TX = T(XYX) for all x E HX. We

shall also identify the adjoined identity in CLO(Y)+ as the identity operator

acting on HY. Let O < 6 < 1/100. For any u E QUaXrXsXk(X)) we can define uxl =

Xx1UXxl, where Xx1 is the characteristic function of X1. Define

1 ( 0 I ) (-UX1 I ) ( O I ) ( I O )

The above formula has its origin in [26]. Let Pl (x) be the Ith Taylor polynomial

for 1/(10A/1-x). Choose lo to be the smallest integer such that

lP10(z) - 1/(107/1 - X)l < b/100

for all x E [O, 99/100]. Notice that

O < I-w1wl/lOO < 99/100.

Let wu = w1 Plo (I-w1 w1 / 100) .

Define

Ao (u) = Xbd1olOr (x1 )nbd1OIOr (x2) wu (I @ °)wuXbdlolOr (xl )nbdlOlOr (x2),

where Xbd1OlOr(xl)nbdlOlOr(x2) is the characteristic function of bdlolor(xl) n

bdlolor (X2)

Let N(6) = lOOlo. Notice that Pl (x) has nonnegative coefficients and

{PI (x)}l is uniformly bounded on [O, 99/100]. This, together with the definition

of Ao(u), implies that there exists a universal constant No such that do(u) E

QpNo6XN(6)rXNosXk(bdN(6)r/loXl n bdN(E)r/10X2) Assume that r < ro/(1 + N(6)), where ro is as in Definition 5.1. Let

0 be the proper strong Lipschitz map from bdy(^)r/lo(xl) n bdN(^)r/lo(x2) to X1 n X2 realizing the strong Lipschitz homotopy equivalence in Definition

5.1; let Vf(t) be the family of isometries as in Lemma 3.3, where {Ek}k (in

Lemma 3.3) is chosen in such a way that SUpk Ek < r/10. We define the boundary of u by:

8(u) = Ad(Vf ) (do (u))-(I @ O) .

Consider the following sequence:

QUE,r,s,k(Xl) @ QUE,r,s,k(X2) QUE,r,s,k(X) GQpNo6XN(6)rXNosXk(xl n X2),

where j(u1 @ u2) = (u1 + Xx-x1 ) (33 (U2 + XX-X2), r < ro/(l + N(6))

NOVIKOV CONJECTURE 345

LEMMA 5.2. Let (X;X1,X2) be as in I)ehinition 5.1. The above sequence is asymptotically exact in the following sense:

(1) For any O < 6 < 1/100,r > O,s > O, there exists O < 61 < 6,0 < r1 < mintr,ro/(l + N(6l))},sl > 0, such that Aj(ul @ u2) is (E,r,sl)-eq?livalent to O for any Ui E QU^lXrlXsXk(xi) (i = 1,2), where 61 depends only on 6 and s; r depends only on 6, r and s; and sl depends only on s;

(2) For any O < 6 < 1/100,r > O,s > O, there exist O < 62 < 6,0 < r2 < minfr,ro/(1 + N(62))}ss2 >- O, such that if u is an element in QU^;2,r2,sXk(x)

for which t3(u) is (62,r2,s)-equivalent to O in GQP^2Xr2XSXk(Xl n x2), then there exists Ui E QUE,r,s2,k(Xi) (i = 1,2) s?lch that j(ul (33 u2) is (E,r,s2)-equivalent to u, where 62 depends only on 6, s, rO and co (co is as in the I)efinition 5.1); r2 depends only on 6, r, s, rO and co; and S2 depends only on s, rO and co.

Proof. (1) follows from the definition of the boundary map. (2) By Lemmas 4.6, 4.9 and the definition of boundary map, for any O <

6' < 6 and O < r2 < mintr, ro/(l +N(62))}, there exist O < 62 < 6' (62 depends only on 6', s and co) and O < r2 < r2 (r2 depends only on 6', r2, s and co) such that, for any u in QUa2Xr2XSXk(X) whose boundary 0(ls) is (62, r2, s)-equivalent to 0, there exists y E QUE/Xcl(6/vs)r2Xc2(s)k(bdy(E2)r2/lo(xl)Xbdy(62)r2/lo(x2)) for which

lixw(I @ O)w*x*-(I @ °)1l < c3(s)6',

where x Y + xx-bdN(62)r2/lo(xl)nbdN(^2)r2/lo(x2)) w = wqbeI and y(62) are as in the definition of the boundary 0(u @ I); cl(E', s) depends only on 6', s and co; c2(s) and C3(5) depend only on s and coe

This implies that

(A) llxw(I (E3 O)-(I @ O)xwll < lOc3(s)E'.

Hence we have

(B) xw= ( d ) ) llbll < lOc3(s)d', llcll < lOc3(s)6t.

Define V1 = aXbdN(62)r2/10(Xl)'

(A) and (B) imply that vl is a b"-quasi-unitary acting on HbdN(^2)r2,lo(xl)) where 6" = 100E'(1 + No)(l + C3(5) + c23(s)) (No is as in the definition of 8(u@I)). (B), together with the definition of w, implies that there exists a constant N1 (E', s) (depending only on 6', s and co) such that

(C) 1l %X-bdNl (^/ ,s)r2 (X2) (U1 (U @ I)-I) || < c4 (s)6' v

(D) {l(Vl (U @ I) I)XX-bdnl(s',S)r2(X2) 1l < C4(s)6,

346 GUOLIANG YU

where C4(S) depends.only on s. We choose

6' < b/101°(l + No)(l + Co + C3(S) + C3(S) + C4(S))v

where. Co is as in Lemma 4.8 (depending only on the Lipschitz constant co in our case). Define

V2 = Xbdnl (6/ S)r2 (x2)vl (U @ I)Xbdnl (6/,s)r2 (X2) *

(C) and (D), together with the choice of 6', imply that v2 is a (6/(1 + Co))- quasi-unitary acting on HbdNl(^/,s)r2(X2)

We have

propagation(vi) < r2(1 + S(62) + Cl (E, s))

We require

O < r2 < ro/lO0(l + N1(E', s) + (62)),

where ro is as in Definition 5.1. Let f1 be the proper strong Lipschitz map from bdN(E2)r2/1o(Xl) to X1 realizing the strong Lipschitz homotopy equivalence in Definition 5.1; let f2 be the proper strong Lipschitz map from bdNl(Eo,s)r2(X2) to X2 realizing the strong Lipschitz homotopy equivalence in Definition 5.1. Define ui = Ad(V1ei)(vi) for i-1,2, where Ad(V1ei) is as in Lemma 3.3, and {6k} (in Lemma 3.3) is chosen in such a way that SUpk Ek < r2.

(C) and (D) imply that (vl +Xx-bdN(452)r2/10 (xl ) ) d3 (U2 +Xx-bdN1 (^/ ,S)r2 (X2) ) is (U$/(1+Co), r3, s3)-equivalent to uG3I, where r3 = C5(6/, s)r2 for some C5(6/, S)

depending only on $', s and co; and s3 depends only on s and co. This, together with the construction of u1 and u2 and Lemma 4.8, implies that u1 and u2 satisfy the desired properties if we choose appropriate r2 and s2, where r2 depends only on $, r, s, ro and co; and s2 depends only on s, ro and co. O

Combining. Lemma 5.2 with Lemma 4.7 we have the following asymptoti- cally exact sequence for QU when k > 1:

QUa,r,s,(Xl) @ QUa,r,s,k(X2) > QUa,r,s,k(X) > QUa,r,s,k_l(xl n X2).

- 6. Spaces with finite asymptotic dimension

In this section we shall prove that the class of finitely generated groups with finite asymptotic dimension (as metric spaces with word length metrics) is hereditary. We shall also prove a localization result which will play an important role in the proof of our main result.

Definition 6.1 ([14]). The asymptotic dimension of a metric space is the smallest integer m such that for any r > O, there exists a uniformly bounded

347 NOVIKOV CONJECTURE

cover C = {Ui}isI of the metric space for which the r-multiplicity of C is at most m + 1; i.e., no ball of radius r in the metric space intersects more than m + 1 members of C.

Notice that the concept of asymptotic dimension is a coarse geometric analogue of the coarering dimension in topology. It is not difficult to lrerify that the asymptotic dimension is inlrariant under quasi-isometry. In particular, this implies that if a finitely generated group is endowed with a word-length metric, then its asymptotic dimension does not depend on the choice of the word-length metric.

PROPOSITION 6.2. Let r be a finitely generated group and rl be a finitely generated subgroqxp of r. If r has finite asymptotic dimension as a metric space with a word-length metric, then rl also has finite asymp-totic dimension as a metric space with a word-length metric.

Proof. For any r > O, let C- {Ui}iEI be the courer for r as in Defi- nition 6.1. Define a colrer C1 for r1 by: C1 = {Ui n r1}iEI Choose finite generating sets G and G1 for r and r1 such that G1 C G, G and G1 are closed under the inlrerse operation. Let d and dl be respectierely the word length metrics on r and r1 associated to G and G1. We hazze d(X,Y) < dl(x,y) for all x, y E r1. This implies that the r-multiplicity of C1 is at most m + 1.

Next we shall prove that C1 is uniformly bounded with respect to d1. By assumption there exists R > O such that the diameter of Ui n rl is at most R with respect to d for all i. This implies that for any x E Ui n rl, we can write X = Xo91 . . . 9k for k < R, where 9l E G for all O < I < k, and xo is a fixed element in Uinr1. Let AR be the set of all elements in r1 which can be written as h1... hj, where h1,..., hj E G, and j < R. It is not difficult to see that AR is a finite set. Therefore there exists R1 > O such that d1(AR,id) < R1. This implies that the diameter of Ui n rl is at most R1 with respect to d1 for all i. [:

Gromolr prolred that hyperbolic groups hazze finite asymptotic dimension as metric spaces with word-length metrics (see page 31-32 in [14]). This result, together with Lemma 6.2, implies that finitely generated subgroups of hyper- bolic groups hazze finite asymptotic dimension as metric spaces with word- length metrics.

Let X be a proper metric space with finite asymptotic dimension m. By the definition of asymptotic dimension there exists a sequence of colrers Ck of X for which there exists a sequence of positive numbers Rk > oo such that (1) Rk+1 > 4Rk for all k; (2) diameter(U) < Rk/4 for all U E Ck; (3) the Rk-multiplicity of Ck+l is at most m Jr 1; i.e., no ball with radius Rk intersects more than m + 1 members of Ckil

348 GUOLIANG YU

Let Ck = {B(U, Rk) I U E Ck+1}, where B(U, Rk) = {x E X I d(x, U) <

Rk}. (1) and (2) imply that {Ck}k is an anti-Cech system for X. Fix a positive integer no. For each n > no, let rn = 7r(&Rn+ -1). By

property (1) of the sequence {Rk}, there exists ni > no such that rn > 7r/2 if n > n1 and there exists a sequence of nonnegative smooth functions {Xn}n>nl on [0,oo) for which (1) Xn(t) = 1 for all 0 < t < 7r/2, and Xn(t) 0 O for all t > rn; (2) there exists a sequence of positive numbers En -O 0 satisfying nX'(t)l <En < 1 for all n > n1.

For each U E Cn+1 (n > n1), define

U' = {V E Nc/0 IV E Cn'o V n u =/ oil

where V E Nc, is the vertex of Nc, corresponding to V E Cn. We define a map Gn from Ncl0 to Ncl by:

Gn (x) = X (dx U) B (U Rn) LTECL+l >EVIECn~l Xn (d(x, V')) BU ~

for all x E N0s. Lemma 6.4 (1) shows that Gn(x) is indeed in Ncn. Let n > n1. We can choose the map inon from Ncn to Ncn in Conjecture

2.7 in such a way that, for each V E Cno+,

inon(B(VI Rno)) = B(U, Rn)

for some U E Cn+1 satisfying U nv v 4 0. Define

F(t, x) = tGn(X) + (1- t)inon(X)

for all t E [0, 1], and x E Ncn/.

LEMMA 6.3. Let X be a proper metric space with finite asymptotic di- mension m, and Gn, F and inon be as above.

(1) Gn is a proper Lipschitz map from NC/n to Ncn with a Lipschitz constant depending only on m;

(2) F(t,x) is a strong Lipschitz homotopy between Gn and inon with a Lipschitz constant depending only on m;

(3) For any e > O,R > O, there exists K > 0 such that d(Gn(x),Gn(y)) <6F

if n > K, d(x,y) < R, where Nco and Ncn are endowed with the spherical metrics.

Proof. (1) By property (1) of Xn)

(A) E Xn(d(x, V')) > 1 VECn+1l

for all x E N0s Cn/o

349 NOVIKOV CONJECTURE

Let U be an element in Cn+l such that Xn(d(z7 U')) + O for some x E

NCro. By property (l) of Xnv

(B) d(x) U/) < rn-

Let x = EtiB(EivRno)v

where ti > °, Ei ti = 1, Vi E CnO+l. Combining (B) with the definition of the

spherical metric on Ncs O, we have

(C) d(Vi, U) < (rn + 7rl2)2Rno+l2/7r + 2Rnoul = Rn/2)

where d(Viv U) is the distance between.two subsets Vi and U of X. (C) implies

that nU Xn(d(xXul))foB(u) t) + 0

Hence Gn(x) is in Nce for all x E NCto. The inequality (C), together with the

condition on the Rn-multiplicity of Cn+l v implies that there are at most m + l

number of nonzero terms in Evcn+1 xn(d(xv V')). Now (1) follows from the

above fact, (A) and property (2) of Xn (2) We shall first prove that, for each x e Nc,ov Gn(x) and inon(z) live

on a common simplex of Ncs . Let

x = E tiB(Viv Rno)v i

where ti > °, Ei ti = l, and Vi e Cnoulh By the choice of inonv

inon (X)-E tiB(Ui E Rn) v

where Ui E Cnl v Ui n vi + 0.

xn(d (xv Ui/)) = 1

for all i. Hence both inon(X) and Gn(x) live on the simplex with vertices

{B(U,Rn) } U E Cn+lvXn(d(zvu/)) 7£ O}, which is a simplex in Nct by the

argument in the proof of (l). Now it is not difficult to see (2). (3) follows from the properties of XnX Gn and an argument similar to that

in the proof of (l). 2

7. Main result and applications

In this section we shall.prove the following result and discuss its applica-

tions to topology, geometry and analysis.

350 GUOLIANG YU

THEOREM 7.1. The coarse Baum-Connes conjecture holds for proper metric spaces with finite asymptotic dimension.

The above theorem greatly improves a result in [40], where, using a con- struction of a Chern character, it is proved that the indices of Dirac type operators on a uniformly contractible Riemannian manifold M are nonzero if the asymptotic dimension of M is equal to the dimension of M.

Combining Theorem 7.1 with the descent principle (Theorem 2.8), we obtain the following corollary.

COROLLARY 7.2. Let r be a finitely generated group whose classifying space has the homotopy type of a finite CW-complex. If rhas finite asymptotic dimension (as a metric space with a word-length metric), then the Novikov conjecture holds for r.

The class of finitely generated groups with finite asymptotic dimension is extremely large and is hereditary (cf. Lemma 6.2). A result of Gromov in [14] and Lemma 6.2 imply that finitely generated subgroups of hyperbolic groups have finite asymptotic dimension. Currently no example of a finitely generated group w;+h infinite asymptotic dimension and finite classifying space is known.

combining Theorem 7.1 with Propositions 4.33 and 4.46 in [31], Theo- rem 3.2 in [38] and the Lichnerowicz argument [25], we have the following application to the positive scalar curvature problem.

COROLLARY 7.3. A uniformly contractible Riemannian manifold with finite asymptotic dimension cannot have uniform positive scalar curvature.

In particular, this corollary implies the Gromov-Lawson-Rosenberg con- jecture for compact K(ir, l)-manifolds when the fundamental group ir has finite asymptotic dimension as a metric space with a word-length metric. Recall that the Gromov-Lawson-Rosenberg conjecture says that there is no Riemannian metric with positive scalar curvature on compact K(7r, 1)-manifolds.

Combining Theorem 7.1 with Propositions 4.33 and 4.46 in [31], and The- orem 3.2 in [38], we have the following application to the spectrum of the Laplacian.

COROLLARY 7.4. Gromov's zero-in-the-spectrum conjecture holds for uniformly contractible Riemannian manifolds with finite asymptotic dimension.

Recall that Gromov's zero-in-the-spectrum conjecture says that the spec- trum of the Laplacian operator acting on the space of L2-forms of a uniformly contractible Riemannian manifold with bounded geometry contains zero ([13], [14]).

Recall that the dimension of a simplicial complex X is the smallest integer m such that each simplex of X has dimension at most m.

351 NOVIKOV CONJECTURE

PROPOSITION 7.5. Let X be a simplicial complex with finite dimension m and endowed with the spherical metric. For any k > m + 1, 0 < 6 < 1/l00,

r > O, s > O, there exist O < 61 < 6, O < r1 < r, s1 > s such that every element u in QUylXrlXsXk(X) is (U$,r,sl)-equivalent to I, where 61 depends only on 6,s, k and m; r1 depends only on r$,r,s,k and m; and sl depends only on s,k, and m.

Proof. Let x(n) be the n-skeleton of X and be endowed with the subspace metric induced from the spherical metric on X. We shall prove our proposition for x(n) by induction on n.

When n = O, we choose rl to be min{r, 7r/10}. By the choice of rl, u has propagation 0. Without loss of generality we can assume that u(t) is piecewise smooth with respect to t E [O, oo) and || ddt || is at most l/c for some constant c > O. For each to E [O, oo), we define

Uto (t) = { u(ct-cto) if cto < t < oo.

Let Hx be as in Definition 2.3. Define

H°X° = (fflkoo=oHx) @ HX.

Notice that H°x° is a standard nondegenerate X-module. Consider

Wl(t ) = (@k=ouk (13 I)(Ifflk-lUk-t' @ I))

where t' E [0,1] and wl(t') acts on H°x°. Notice that

Wl(0) = U ek°°=l I (13 I.

It is easy to see that there exists a smooth homotopy v(t') (t' E [0,1]) such that

t(O) = Ifflk°°=luk 11 @ I;

v(l) = fflk=oUk 1 (33 I;

llv/(t/)ll ' 100

for all t' E [0,1]. Define

(t') _ X wl(2t') if O < t' < 1/2; l (fflk=0ak 033 I)v(2t' - 1) if 1/2 < t' < 1.

It is not difficult to see that w(t') realizes the (U$,r,sl)-equivalence between u @ I and I if we choose appropriate r$1 and sl, where r$1 depends only on r$, and sl depends only on s.

Assume by induction that the proposition holds when n = I-1. Next we shall prove the proposition when n = 1. For each simplex A of dimension I in X, we define

A1 = {X E A | d(X,C(t)) < 1/l00}, 2 = {X E t } d(x,c(A)) > 1/100},

352 GUOLIANG YU

where c(A) is the center of A. Let

X1 = U A: simplex of dimension I in X Ai X2 = UA: simplex of dimension 1 in XA2.

Notice that (1) Xi is strongly Lipschitz homotopy equivalent to

{c(A) j A: l - dimensional simplex in X};

(2) X2 is strongly Lipschitz homotopy equivalent to X(1-1); (3) XMl) = X1 U X2, and X1 f x2 is the disjoint union of the boundaries

of all i-dimensional A1 in X(l). (1) and (2), together with Lemma 4.8 and the induction hypothesis, imply

that our proposition holds for X1 and X2. By Lemmas 4.8, 4.7 and 5.2, we also know that our proposition holds for Xi n X2. It is not difficult to verify that (X(l); X1, X2) satisfies the strong excision condition in Definition 5.1. Now we can complete the induction process by using the asymptotically exact sequence for QU (cf. Lemmas 5.2 and 4.7). D-

Proof of Theorem 7.1. Let X be a proper metric space with asymptotic dimension m. By Theorem 3.5 it is enough to prove that

lim Ki (CZ,O (Nc )) = 0, n--+oo

)

where Cn7 is as in Section 6. By Lemma 4.5 any element in Ki(CZ0 (NCn )) can be represented as an element u C QU61,r,s,k (Nc/no) for some r, s and k > m +1, where 61 is as in Proposition 7.5 for some 0 < 6 < 1/100. Let

Un = Ad(VGn)(U)v

where Gn is as in Lemma 6.3, Ad(VGn) is as in Lemma 3.3, and {ek}k (in Lemma 3.3) is chosen in such a way that supkek < r1/10, where ri is as in Proposition 7.5. By Lemma 6.3, there exists K > 0 such that un has propa- gation at most ri for n > K. Our assumption on the asymptotic dimension of X implies that the dimension of Ncn is at most m for all n. By Proposition 7.5 we know that un is (6, r, si)-equivalent to I in QU6,r,s,k(NCn) for n > K, where sl is as in Proposition 7.5. By Lemmas 4.8 and 6.3, un is equivalent to Ad(Vinon) (u) in Ki (CL,O (Ncn/)). Hence [u] = 0 in limnoo Ki (CL, (NC)) D

8. A counterexample to the coarse Baum-Connes conjecture

In this section we shall construct a proper metric space with infinite asymptotic dimension for which the coarse Baum-Connes conjecture fails. This indicates that our result on the coarse Baum-Connes conjecture is best possible in some sense.

NOVIKOV CONJECTURE 353

Let X be the disjoint union of S2n (n = 1, 2,3,...), where S2n is the sphere of dimension 2n. Endow a metric d on X such that

(1) dIs2n= ndn, where d s2n is the restriction of d to S2n, and dn is the standard Riemannian metric on the sphere S2n with radius 1;

(2) d(S2n, S2n') > max{n, n'} if n 74 n'.

PROPOSITION 8.1. Let X be the proper metric space defined as above. (1) X is a counterexample to the coarse Baum-Connes conjecture; (2) X has infinite asymptotic dimension.

Proof. Let Dn be the Dirac operator on S2n. Define

D = _

D gives rise to a K-homology class [D] in Ko (X). Let kn be the scalar curvature for S2n with the Riemannian metric ndn. We have

2n(2n - 1) 1 kn =

2 p2 -

This, together with the Lichnerowicz formula, implies that D is invertible. It follows that the index of D, Ind(D), is zero in Ko(C*(X)).

Let {Ck}21I be an anti-Cech system for X such that each Ck is an open cover for X. Let {diu}uEC, be a continuous partition of unity subordinate to C1. Define a proper continuous map q$ from X to NC1 by:

(x) = E qu(x)U. UEC1

By the properties of the metric d on X, it is not difficult to see that 40* ([D]) 7 0 in limk oo Ko(Nck). This, together with the fact that Ind(D) = 0, implies that X is a counterexample to the coarse Baum-Connes conjecture.

The proof for (2) is straightforward and is therefore omitted.

Finally we remark that, based on the above example, one can construct a counterexample to the coarse Baum-Connes conjecture which is a connected finite-dimensional Riemannian manifold and has infinite asymptotic dimension.

UNIVERSITY OF COLORADO, BOULDER, CO Email address: GYU@EUCLID.COLORADO.EDU

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(Received September 12, 1995)