MODEL THEORETIC REFORMULATION OF THE BAUM-CONNES AND FARRELL-JONES CONJECTURES

PAUL BALMER AND MICHEL MATTHEY

Abstract. The Isomorphism Conjectures are translated into the language of homotopical algebra, where they resemble Thomason’s descent theorems.

1. Introduction and statement of the results

In [8], Thomason establishes that algebraic K-theory satisfies Zariski and Nisnevich descent. This is now considered a profound algebraico-geometric property of K-theory. In [1, 2], we have introduced the sister notion of codescent. Here, we prove that each one of the so-called Isomorphism Conjectures (see [3, 5]) among

(1) the Baum-Connes Conjecture,

(2) the real Baum-Connes Conjecture,

(3) the Bost Conjecture,

(4) the Farrell-Jones Conjecture in K-theory,

(5) the Farrell-Jones Conjecture in L-theory, is equivalent to the codescent property for a suitable K- or L-theory functor.

For a (discrete) group G, these conjectures aim at computing, in geometrical and topological terms, the groups K_{∗}^{top}(C_{r}^{∗}G), KO_{∗}^{top}(_{ℝ}C_{r}^{∗}G), K_{∗}^{top}(ℓ^{1}G), K_{∗}^{alg}(RG) and L∗^{⟨−∞⟩}(ΛG) respectively, where R and Λ are associative rings with units, and Λ is equipped with an involution. Davis and Lück [4] express these conjectures as follows (the equivalence with the original statements is due to Hambleton-Pedersen [6]). First, fix one of the Conjectures (1)–(5) and denote by K_{∗}(G) the corresponding Kor L-group among the five listed above (for (4) and (5), R and Λ are understood). Denote by C := Or(G) the orbit category of G, whose objects are the quotients G/H with H running among the subgroups of G, and the morphisms are the left-G-maps. Let D := Or(G, VC) be the full subcategory of Or(G) on those objects G/H for which H is virtually cyclic. We sometimes write C_{G} and D_{G} to stress the dependence on the group G. Then, a suitable functor X_{G}: C ⟶ S is constructed, where S denotes the usual stable model category of spectra (of compactly generated Hausdorff spaces), for which the weak equivalences are the stable ones. This functor X_{G} has the property that π_{∗}(X_{G}(G/H)) is canonically isomorphic to K_{∗}(H) for all H ⩽ G. Then, the fixed Isomorphism Conjecture for G amounts to the statement that the following composition, called assembly map, is a weak equivalence in S:

μ^{G}: hocolim_{D} res_{D}^{C} X_{G} ⟶ hocolim_{C} X_{G} ⟶^{∼} colim_{C} X_{G} ⟶^{≅} X_{G}(G/G) .

We turn to homotopical algebra. First, we denote by U_{S}(C, D) the model category S^{C} of functors C → S, where the weak equivalences and fibrations are the D-weak equivalences and D-fibrations respectively, i.e. they are defined D-objectwise. See details in [1, § 3], for instance. For a diagram X ∈ S^{C}, we let ξ_{X}: QX ⟶ X be the cofibrant replacement of X in U_{S}(C, D). As in [1, § 4], we say that X satisfies Dcodescent if the map ξ_{X}(c) is a weak equivalence in S for every c ∈ C; if this is only fulfilled at some c_{0} ∈ C, we say that X satisfies D-codescent at c_{0}. For a conceptual approach to codescent and a parallel with descent, see [1, §§ 1 and 5]. Let U_{S}(C) be the model structure on S^{C} with the C-weak equivalences and C-fibrations; we define U_{S}(D) on S^{D} similarly. We denote by Ho_{S}(C) and Ho_{S}(D) the homotopy category of U_{S}(C) and U_{S}(D) respectively. As in [1, Prop. 13.2], we have the derived adjunction of the Quillen adjunction ind_{D}^{C}: U(D) \overrightarrow{⟵} U(C) : res_{D}^{C}, namely

Date: August 6, 2003.

Research supported by Swiss National Science Foundation, grant 620-66065.01.

Lind_{D}^{C}: Ho_{S}(D) \overrightarrow{⟵} Ho_{S}(C) : Res_{D}^{C} .

For the sequel, fix a group G and one of the Isomorphism Conjectures (1)–(5); let X_{G} ∈ S^{C} be the corresponding functor. Keep the other notations as above.

Theorem 1.1. The following statements are equivalent :

(i) G satisfies the considered Isomorphism Conjecture;

(ii) the corresponding functor X_{G} ∈ U_{S}(C, D) satisfies D-codescent at G/G ∈ C.

Theorem 1.2. For subgroups L ⩽ H ⩽ G, the following statements are equivalent : (i) X_{H} ∈ U_{S}(C_{H}, D_{H}) satisfies D_{H}-codescent at H/L ∈ C_{H}; (ii) X_{G} ∈ U_{S}(C_{G}, D_{G}) satisfies D_{G}-codescent at G/L ∈ C_{G}.

In fact, by general results of [1] (without invoking 1.1 above), if X_{G} satisfies D_{G}-codescent, then X_{H} satisfies D_{H}-codescent for every subgroup H ⩽ G.

Main Theorem. The following statements are equivalent :

(i) every subgroup H of G satisfies the considered Isomorphism Conjecture; (ii) the corresponding functor X_{G} ∈ U_{S}(C, D) satisfies D-codescent; (iii) up to isomorphism, the image of X_{G} in Ho_{S}(C) belongs to Lind_{D}^{C} ( Ho_{S}(D)).

Note that the usual Baum-Connes and Bost Conjectures are stated with finite subgroups instead of virtually cyclic ones, but this is known to be equivalent. So, in these cases, we could as well set D_{G} := Or(G, Fin) instead of Or(G, VC).

Remark 1.3. Let X ∈ S^{C} be a diagram and let ζ_{X}: QX ⟶ X be an arbitrary cofibrant approximation of X in U_{S}(C, D), namely, ζ_{X} is merely a D-weak equivalence and QX is cofibrant in U_{S}(C, D). Then, X satisfies D-codescent at some object c ∈ C if and only if ζ_{X}(c) is a weak equivalence in S, see [1, Prop. 6.5]. This illustrates the flexibility of the codescent-type reformulation of the Isomorphism Conjectures, namely, every such cofibrant approximation of X_{G} yields a possibly very different assembly map that can be used to test the considered conjecture.

2. The proofs

Let Gpds^{f} be the category of groupoids with faithful functors. For the considered conjecture, by [4, 7], there exists a homotopy functor X : Gpds^{f} ⟶ S, i.e. X takes equivalences of groupoids to weak equivalences, such that X_{G} is the composite

X_{G}: C = Or(G) ⟶^{ι} Gpds^{f} ⟶^{X} S.

The functor ι takes G/H to its G-transport groupoid \overline{G/H}^{G} with the set G/H as objects and with {g ∈ G | gg_{1}H = g_{2}H } as morphisms from g_{1}H to g_{2}H. Moreover, the functor X takes values in cofibrant spectra, so that X_{G} is C-objectwise cofibrant.

Let Cat be the category of small categories and sSets that of simplicial sets. Denote by ⊗_{D}: sSets^{D^{op}} × S^{D} ⟶ S the tensor product over D induced by the simplicial model structure on S, where K ∈ sSets “acts” on E ∈ S by |K|_{+} ∧ E. Proof of Theorem 1.1. A priori, to test whether X_{G} satisfies D-codescent at some c ∈ C requires a thorough understanding of the usually mysterious cofibrant replacement of X_{G}. A key point here is the freedom to use any cofibrant approximation instead, see Remark 1.3. We provide in [2, § 6] a general construction of cofibrant approximations in U_{S}(C, D), one of which is exactly suited for our present purposes [2, Cor. 6.9]. Evaluated at the terminal object G/G ∈ C, this cofibrant approximation ζ_{X_{G}}: QX_{G} ⟶ X_{G} is a certain map (described at the end of the proof)

ζ_{X_{G}}(G/G): QX_{G}(G/G) = B(?↘D)^{op} ⊗_{?∈D} res_{D}^{C} X_{G}(?) ⟶ X_{G}(G/G) .

Indeed, using the notations of [2, Not. 6.1], this follows from the canonical identification (? ↘ D ↘^{C} G/G)^{op} = (? ↘ D)^{op} of diagrams in Cat^{D^{op}} and from the fact that X_{G} is C-objectwise cofibrant. By definition of the homotopy colimit, we have

B(?↘D)^{op} ⊗_{?∈D} res_{D}^{C} X_{G}(?) = hocolim_{D} res_{D}^{C} X_{G} .

So, it suffices to show that ζ_{X_{G}}(G/G) coincides with the assembly map μ^{G}. In the notations of [2, Not. 5.1], we have \underline{mor_{D,C}(?, G/G)} = ∗ in sSets^{D^{op}} (the constant diagram with value the point). By [2, Lem. 5.3], the spectrum ∗ ⊗_{D} res_{D}^{C} X_{G} identifies with ind_{D}^{C} res_{D}^{C} X_{G}(G/G). Letting ϵ denote the counit of the adjunction (ind_{D}^{C}, res_{D}^{C}), it is routine to verify that there is a canonical commutative diagram

[figure: hocolim_{D} res_{D}^{C} X_{G}; B(?↘D)^{op} ⊗_{D} res_{D}^{C} X_{G}(?); hocolim_{D} res_{D}^{C} X_{G}; colim_{D} res_{D}^{C} X_{G}; ≅; ∗ ⊗_{D} res_{D}^{C} X_{G}; ϵ_{X_{G}} (G/G); hocolim_{C} X_{G}; ∼; colim_{C} X_{G}; ≅; X_{G}(G/G)]

The composition of the first column followed by the last row is the assembly map μ^{G}. The composition in the last column is ζ_{X_{G}}(G/G), see [2, Cor. 6.9]. □

More generally, one can prove that the “ (X, F, G)-Isomorphism Conjecture ” of [4, Def. 5.1] is equivalent to X satisfying Or(G, F)-codescent at G/G, for any objectwise cofibrant diagram X ∈ S^{Or(G)} and any family F of subgroups of G.

For g ∈ G and H ⩽ G, we write^{g}H := gHg^{−1}. In the orbit category Or(G) = C_{G}, for an element g ∈ G such that ^{g}H ⩽ K for some subgroups H and K of G, we designate by the right coset Kg the morphism G/H ⟶ G/K taking g̃H to g̃g^{−1}K.

Proof of Theorem 1.2. Consider the functor Φ: C_{H} ⟶ C_{G} taking a coset H/L ∈ C_{H} to G/L. For any L ⩽ H, we have canonical equivalences of groupoids in Gpds^{f}

\overline{H/L}^{H} ⟵^{∼} \overline{L} ⟶^{∼} \overline{G/L}^{G} ,

where \overline{L} is L viewed as a one-object groupoid. Since X is a homotopy functor, one checks that there is a canonical zig-zag of two C_{H}-weak equivalences between X_{H} and Φ^{∗}X_{G} = X_{G} ∘ Φ in U_{S}(C_{H}). By weak invariance of codescent [1, Prop. 6.10], X_{H} and Φ^{∗}X_{G} satisfy D_{H}-codescent at exactly the same objects H/L of C_{H}.

Fix an object H/K ∈ D_{H}. Let E_{H/K} ⊂ G be a set of representatives for the quotient H\{g ∈ G |^{g}K ⩽ H}. Let Mγ: Φ(H/K) = G/K ⟶ G/M = Φ(H/M) be a morphism in C_{G} with M ⩽ H (and γ ∈ G). It is straightforward that there is a unique pair (g, Mh) with g ∈ E_{H/K} and Mh ∈ mor_{C_{H}} (H/^{g}K, H/M) (namely characterized by Hg = Hγ and Mh = Mγg^{−1}) such that Mγ decomposes in C_{G} as

\xymatrix{G/K \ar[r]^{^{g}Kg} \ar[rr]^{Mγ} & G/^{g}K \ar[r]^{Mh} & G/M .}

Since Φ(D_{H}) ⊂ D_{G}, this precisely says that Φ is a left glossy morphism of pairs of small categories in the sense of [1, Defs. 7.3 and 8.1]. By left glossy invariance of codescent [1, Thm. 9.14], Φ^{∗}X_{G} satisfies D_{H}-codescent at some H/L ∈ C_{H} if and only if X_{G} satisfies D_{1}-codescent at G/L ∈ C_{G}, where D_{1} := Φ(D_{H}). Set D_{2} := D_{G} and fix H/L ∈ C_{H}. For i = 1, 2, consider the full subcategory E_{i} of D_{i} given by

E_{i} := {G/K ∈ D_{i} | mor_{C_{G}}(G/K, G/L) ≠ ∅} .

By the Pruning Lemma [1, Thm. 11.5], X_{G} satisfies D_{1}-codescent at G/L if and only if it satisfies E_{1}-codescent at G/L. Since L ⩽ H, every object of E_{1} is isomorphic, inside C_{G}, to some object of E_{2} and conversely; in other words, E_{1} and E_{2} are essentially equivalent in C_{G}, in the sense of [1, Def. 3.12]. So, by [1, Prop. 10.1], X_{G} satisfies E_{1}-codescent at G/L if and only if it satisfies E_{2}-codescent at G/L. By the Pruning Lemma again, X_{G} satisfies E_{2}-codescent at G/L if and only if it satisfies D_{2}-codescent at G/L, i.e. D_{G}-codescent at G/L.

In total, we have proven that X_{H} satisfies D_{H}-codescent at an object H/L ∈ C_{H} if and only if X_{G} satisfies D_{G}-codescent at G/L, as was to be shown. □ Proof of the Main Theorem. The equivalence between (i) and (ii) follows from Theorems 1.1 and 1.2; (ii) and (iii) are equivalent by [1, Thm. 13.5]. □

References

[1] P. Balmer and M. Matthey. Codescent theory I : Foundations, Preprint, 2003.

[2] P. Balmer and M. Matthey, Codescent theory II : Cofibrant approximations, Preprint, 2003.

[3] P. Baum, A. Connes, and N. Higson. Classifying spaces for proper actions and K-theory of group C^{∗}-algebras, In C^{∗}-algebras : 1943-1993, Contemp. Math. 167, 241–291, 1994.

[4] J. F. Davis and W. Lück. Spaces over a category and assembly maps in isomorphism conjectures in K- and L-theory, K-theory 15 (1998), 201–252.

[5] F. T. Farrell and L. E. Jones. Isomorphism conjectures in algebraic K-theory, J. Amer. Math. Soc. 6 (1993), 249–297.

[6] I. Hambleton and E. K. Pedersen. Identifying assembly maps in K- and L-theory, Preprint (2002), to appear in Math. Ann.

[7] M. Joachim. K-homology of C^{∗}-categories and symmetric spectra representing K-homology, Preprint (2002), to appear in Math. Ann.

[8] R. W. Thomason and T. Trobaugh. Higher algebraic K-theory of schemes and of derived categories. In The Grothendieck Festschrift III, 247–435, Progr. Math. 88, Birkhäuser, 1990.

Department of Mathematics, ETH Zentrum, CH-8092 Zürich, Switzerland

   E-mail address: paul.balmer@math.ethz.ch and michel.matthey@math.ethz.ch
   URL: http://www.math.ethz.ch/∼balmer and http://www.math.ethz.ch/∼matthey