«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 ...»
An Etale Approach to the Novikov Conjecture
University of Florida, Gainesville, FL 32611-8105, USA
Department of Mathematics, Rutgers University, Piscataway, NJ 08854-8019
Department of Mathematics, The University of Chicago, Chicago, IL 60637
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  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  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 .) 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  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 : 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, , 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  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 ). For a subset V ⊂ X we denote by A(V ) the sum ⊕x∈V A(x). Ranicki  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) .
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 .
The ﬁrst part of it is discussed in a more general setting in , Theorem 3.3.
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.
The ﬁnite dimensionality condition is essential here. There are acyclic compacta that have nontrivial mod p complex K-theory . 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 . 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).
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  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 ). We introduce coefﬁcients to the second diagram of Theorem 2.2 by forming the smash product with M(pk ).
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.
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 .
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.