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«Abstract We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson ...»

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An Etale Approach to the Novikov Conjecture

A.N. DRANISHNIKOV

University of Florida, Gainesville, FL 32611-8105, USA

S. FERRY

Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019

AND

S. WEINBERGER

Department of Mathematics, The University of Chicago, Chicago, IL 60637

Abstract

We show that the rational Novikov conjecture for a group Γ of finite homological type follows from the mod 2 acyclicity of the Higson compactifcation of an EΓ. We then show that for groups of finite asymptotic dimension the Higson compactification is mod p acyclic for all p, and deduce the integral Novikov conjecture for these groups. c 2000 Wiley Periodicals, Inc.

1 Introduction Ten years ago, the most popular approach to the Novikov conjecture went via compactifications. If a compact aspherical manifold, say, had a universal cover which suitably equivariantly compactified, already Farrell and Hsiang [20] proved that the Novikov conjecture follows. Subsequent work by many authors weakened the various hypotheses and extended the idea to other settings. (See e.g. [9, 22, 37].) In recent years, other coarse methods (most notably the embedding method of [41] in the C*-algebra setting) have supplanted the compactification method. The reason for this is that the embedding method seemed to have a better chance of applying generally, since compactifications effective for the Novikov conjecture seem to require some special geometry for their construction. For a brief period, it seemed that a general construction might solve the Novikov conjecture. Higson introduced a general compactification of metric spaces (somewhat reminiscent of the Stoneˇ Cech compactification) that automatically has most of the needed properties for application to the Novikov conjecture. The Novikov conjecture would follow if it could be proven that the Higson compactification of the universal cover of an aspherical polyhedron is acyclic. (See [37].) Communications on Pure and Applied Mathematics, Vol. 000, 0001–0018 (2000) c 2000 Wiley Periodicals, Inc.

2 A. N. DRANISHNIKOV, S. C. FERRY, S. A. WEINBERGER Unfortunately, it was soon realized by Keesling and others (see [29, 15]) that the Higson compactification, even for manifolds as small as R, has nontrivial rational cohomology. Thus, it was felt that such universal compactifications were not suitable for Novikov. Recently, however, Gromov has harpooned the embedding approach by constructing finitely generated groups that do not uniformly embed in Hilbert space [23] and a number of authors (e.g. [25, 32]) have shown that Gromovs groups can be used to construct counterexamples to general forms of the Baum-Connes conjecture. Currently, however, there are no candidates for counterexamples to the Novikov conjecture nor is there a potential method for proving the conjecture.

Moreover, for reasons that are not entirely clear, the embedding method has never been translated into pure topology; the results on integral Novikov conjecture so obtained, from the L-theoretic viewpoint never account for the prime 2, and there do not seem to be many results in pure algebraic K-theory (or A-theory) provable by this method.

This paper seeks to rehabilitate the approach to the Novikov conjecture via the Higson compactification. We do this by examining the Higson compactification with finite coefficients.

Theorem 1. If the Higson compactification of Eπ is mod 2 acyclic, and Bπ is a finite complex, then the integral Novikov conjecture for π holds at the prime 2.

In other words, the assembly map localized at the prime two is an injection on homotopy groups. At odd primes we do not know how to prove any corresponding statement, in general, for reasons related to the examples in [16]: The L-spectrum away from 2 is periodic K-theory, and cohomologically acyclic spaces may not be acyclic for periodic K-theory.

This difficulty is an infinite dimensional phenomenon, so for groups of finite asymptotic dimension this issue does not arise, and it is possible to prove integral results.

Theorem 2. If Bπ is a finite complex and π has finite asymptotic dimension, then if R is a ring such that L(R) is finitely generated, then the L(R) assembly map for π is injective.

In this paper, by L(R) we will always mean L−∞ (R).

–  –  –

novelty here is the method of using finite primes to regain the acyclicity of the Higson compactification and then using the etale idea to get our Novikov result.1 Theorem 2 follows from the general discussion above, together with our third theorem.

Theorem 3. If Bπ is a finite complex and π has finite asymptotic dimension, then the Higson compactification of Eπ is mod p acyclic for every prime p.

–  –  –

for any fixed r and any sequence of points x going to infinity in X. Higson’s compactification is the smallest one containing X densely so that all bounded continuous functions with decaying variation extend. Let Ch be the set of such functions.

We obtain the Higson compactification of X by embedding X into ∏ f ∈Ch R and taking the closure of the image. This closure is compact by the Tychonoff theorem, since the maps f are bounded.





Let us now consider the first cohomology of the compactification of a uniformly ˇ¯ ¯ simply connected X. We represent an element of H 1 (X) by a map f : X → S1. On X, we can lift f |X and view our class as being given by exp(2πi f˜), where f˜ has ¯ decaying variation but is not necessarily bounded. Maps f, g : X → S1 represent ˜ − g is bounded.

the same cohomology class iff f ˜ Note that H 1 (X; Z) is now an R-vector space – one can multiply the function f˜ ¯ ¯ by a real number, project back to S1, and extend to X. In particular, this cohomology vanishes with mod p coefficients. In this regard, the cohomology of the Higson compactification resembles the uniformly finite homology of a uniformly contractible n manifold, [4], which is known to be an R vector space, aside from the class coming from the fundamental class of X.

Our paper is organized as follows. Section 2 is devoted to descent, i.e. deducing Novikov conjectures from metric results on the universal cover. The last section is devoted to verifying the mod p acyclicity result for the finite asymptotic dimension case. This is done using ideas of quantitative algebraic topology. The key idea is that when homotopy groups are finite, it is sometimes possible to get extra a priori Lipschitz conditions on maps.

1 Note that the celebrated paper of [5] on the algebraic K-theory analogue of the Novikov conjecture is a very nice illustration of a completely different etale idea. We thank Alain Connes for suggesting this evocative description of our method.

4 A. N. DRANISHNIKOV, S. C. FERRY, S. A. WEINBERGER 2 L-theory and assembly map with Z p coefficients For every ring with involution R (even for any additive category with involution) Ranicki defines a 4-periodic spectrum L∗ (R) with πi (L∗ (R)) = Li (R) [34, 35]. We use the notation L = L∗ (Z). Strictly speaking, L-spectra are indexed by K-groups.

Our L(R) equals L−∞ (R), the limit of the L-spectra associated to the negative K-groups. This applies to all L-spectra occurring in this paper.

For a metric space (X, d) we will denote by CX (R) the boundedly controlled Pedersen-Weibel category whose objects are locally finite direct sums A = ⊕x∈X A(x) of finite dimensional free R-modules and whose morphisms are given by matrices with bounded propagations (see [21]). For a subset V ⊂ X we denote by A(V ) the sum ⊕x∈V A(x). Ranicki [35] defined X-bounded quadratic L-groups L∗ (CX (R)) and the corresponding spectrum L∗ (CX (R)). We will use the notation Lbdd (X) = L∗ (CX (Z)).

¯ ¯ Suppose that X is a compactification of X with corona Y = X \ X. Then one can define the continuously controlled category BX,Y (R) by taking the same objects as above with morphisms f : A → B in BX,Y (R) taken to be homomorphisms such that ¯ for every y ∈ Y and every neighborhood y ∈ U ⊂ X there is a smaller neighborhood y ∈ V ⊂ U such that f (A(V )) ⊂ A(U). This category is additive and hence also admits an L-theory. The corresponding spectrum for R = Z we denote by Lcc (X) = L∗ (BX,Y (Z)).

–  –  –

where x0 ∈ X is any basepoint, Br (x0 ) is the r-ball centered at x0, and the distance d(A, B ) between sets is the infimum of distances between d(a, b), a ∈ A, b ∈ B.

We note that the minimal compactification of a proper metric space X with respect to the property that the closures of every pair of diverging subsets in X have empty intersection in the corona is the Higson compactification.

AN ETALE APPROACH TO THE NOVIKOV CONJECTURE 5

–  –  –

If X is the universal covering of a finite complex with fundamental group Γ, then Γ acts on CX (R) and hence on L∗ (CX (R)) with fixed set L∗ (CX (R))Γ = L∗ (RΓ) [9].

If a compactification of X with corona Y is equivariant, then Γ acts on BX,Y (R) and hence on L∗ (BX,Y (R)).

We recall that the homotopy fixed set X hΓ of a pointed space X with a Γ action on it is defined as the space of equivariant maps MapΓ (EΓ+, X).

For a general spectrum E and a torsion free group Γ it was proven [6, 9] that

–  –  –

The following theorem is due to Carlsson-Pedersen [9, 10]. See also Ranicki [36].

The first part of it is discussed in a more general setting in [39], Theorem 3.3.

Theorem 2.2.

For every group Γ with finite classifying complex BΓ there is a morphism of spectra called the bounded control assembly map

–  –  –

where A is the standard assembly map, and the vertical arrows are the natural maps from fixed sets to homotopy fixed sets.

6 A. N. DRANISHNIKOV, S. C. FERRY, S. A. WEINBERGER If the universal covering space EΓ admits a Higson dominated equivariant compactification X with corona Y, then the diagram can be extended

–  –  –

˜ which converges to H∗ (X; S) provided that X is a finite dimensional compact metric space [19, 28]. Since all of the groups Hq (S0, L ∧ M(pk )) are p-primary, we have Ei, j = 0 for all i and j.

Remark 2.4.

The finite dimensionality condition is essential here. There are acyclic compacta that have nontrivial mod p complex K-theory [42]. Namely, by results of Adams and Toda there is a map f : Σk M(Z p, m) → M(Z p, m) which induces an isomorphism in K-theory. The inverse limit X of suspensions of f is a an acyclic compactum, since all bonding maps are trivial in cohomology, yet it has nontrivial K-theory and nontrivial mod p K-theory. Hence, for p odd, X has nontrivial mod p L-theory.

AN ETALE APPROACH TO THE NOVIKOV CONJECTURE 7

–  –  –

Indeed, for any R, L(R) is Eilenberg-MacLane at 2 [43]. Thus, if a compactum X is Z2 -acyclic, then it is L ∧ M(2)-acyclic without any finite dimensionality assumption on X.

Definition 2.6. The mod p assembly map A p is A ∧ 1M(p) : H∗ (BΓ; L) ∧ M(p) → L∗ (ZΓ) ∧ M(p).

Theorem 2.7.

Suppose that the Higson compactification of the universal cover X of a finite aspherical complex B is finite-dimensional and acyclic in Steenrod mod p homology. Then the mod pk assembly map for Γ = π1 (B) is a split monomorphism for every k.

Proof. Let let m be the dimension of the Higson corona dim νX. Using Schepin’s spectral theorem [12] one can obtain a metrizable Z p -acyclic Γ-equivariant com¯ pactification X of X with corona Y such that dimY = m (see the proof of Lemma

8.3. in [12]). We introduce coefficients to the second diagram of Theorem 2.2 by forming the smash product with M(pk ).

–  –  –

Theorem 2.9.

Suppose that the the universal cover of a finite complex BΓ has finite dimensional Higson compactification which is mod p acyclic for every prime p. Then the integral assembly map A is a monomorphism.

Proof. In view of compactness of BΓ we have νΓ = νEΓ. By Theorem 2.7 A ⊗ 1G is a split monomorphism for every finite abelian group G. Since BΓ is a finite complex, the standard induction argument on the number of cells show that the group Hi (BΓ; L) is finitely generated for every i. Hence for every α ∈ Hi (BΓ; L) there is p such that α ⊗ 1 ∈ Hi (BΓ; L) ⊗ Z pk is not zero. By the UCF there is a

AN ETALE APPROACH TO THE NOVIKOV CONJECTURE 9

–  –  –

This diagram implies that A(α) = 0.

Theorem 2.10.

Suppose that Γ is a group with BΓ a finite complex. If EΓ has mod 2 acyclic Higson compactification, then the rational Novikov conjecture holds for Γ.

–  –  –

Since the group Hi (BΓ; L) is finitely generated, it can be presented as ⊕Fi Z ⊕ Tori.

Since A∗ ⊗ 1Z2k is a monomorphism, the kernel ker(A∗ ) consists of 2k divisible elements. Since k is arbitrary, ker(A∗ |⊕Z ) = 0. Therefore, A∗ ⊗ 1Q is a monomorphism.

This proves Theorem 1 of the introduction.

–  –  –

the minimal Lipschitz constant of f.

Every simplicial complex K carries a metric where all simplexes are isometric to the standard euclidean simplex. We will call the maximal such metric on K uniform 10 A. N. DRANISHNIKOV, S. C. FERRY, S. A. WEINBERGER and usually we will denote the corresponding metric space by KU. 2 Note that the metric space KU is geodesic. If no metric is specified, we will assume that a finite complex is supplied with this uniform metric. In particular, the complexes in the two lemmas below are assumed to have the uniform metric.

The following lemma is a special case of Theorem A from [40].

Lemma 3.1.

Let Y be a finite simplicial complex with πn (Y ) finite. Then for every λ 0 there is a µ 0 such that every map f : Bn → Y with Lip( f |Sn−1 ) ≤ λ can be deformed to a µ-Lipschitz map g : Bn → Y by means of a homotopy ht : Bn → Y with ht |Sn−1 = f |Sn−1.



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