Assembly maps with coefficients in topological algebras and the integral K-theoretic Novikov conjecture
We prove that any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the integral algebraic K-theoretic (split) Novikov conjecture over and , where denotes the -algebra of compact operators and denotes the algebra of Schatten class operators. We introduce assembly maps with finite coefficients and under an additional hypothesis, we prove that such a group also satisfies the algebraic K-theoretic Novikov conjecture over and with finite coefficients. For all torsion free Gromov hyperbolic groups , we demonstrate that the canonical algebra homomorphism induces an isomorphism between their algebraic K-theory groups.
Key words and phrases:K-theory, Novikov conjecture, homotopy groups with coefficients, Baum–Connes conjecture
2010 Mathematics Subject Classification:19D50, 19K35, 46L80
For any discrete group and a unital ring the algebraic K-theoretic Novikov conjecture for over asserts that a canonically defined Loday assembly map
is rationally injective. Here denotes the nonconnective algebraic K-theory spectrum of . In fact, there is a map of spectra , which induces . The stronger integral K-theoretic (resp. split K-theoretic) Novikov conjecture asserts that is injective (resp. split injective). Using standard excision arguments and the fact that H-unital -algebras in the sense of [Wod] satisfy excision in algebraic K-theory [SusWod2], the Loday assembly map can be extended to H-unital coefficient -algebras . Throughout this article the term K-theory without any adjective will refer to algebraic K-theory and will be denoted by . Let us introduce a few more notations.
Notations and conventions:
Loday assembly map with coefficients in ,
Davis–Lück assembly map with coefficients in ,
Baum–Connes assembly map with coefficients in (separable -algebra).
Sometimes we are going to suppress the coefficient algebra or from the notation if it is the complex numbers, i.e., , where . All groups are assumed to be countable and will denote the minimal -tensor product. We are going to freely use the notations in [DavLue].
In this article we prove the following two results (reduction principles):
(see Theorem LABEL:cpt and Theorem LABEL:overS) If a countable discrete and torsion free group satisfies the integral -theoretic Novikov (resp. the split -theoretic Novikov) conjecture with -coefficients, i.e., is injective (resp. split injective), then it satisfies the integral K-theoretic Novikov (resp. the split K-theoretic Novikov) conjecture over and , where denotes the -algebra of compact operators and denotes the algebra of Schatten class operators.
If is merely rationally injective, then so is . (The rational injectivity of is known for all groups without any further hypothesis [GuoliangNovikov]).
(see Theorem LABEL:overC) Let be a countable discrete and torsion free group satisfying the split -theoretic Novikov conjecture with -coefficients, i.e., is split injective. Assume, in addition, that the -homology of is concentrated in even degrees. Then satisfies the K-theoretic Novikov conjecture over and with finite -coefficients.
In the proof of the above Theorem 0.2 one needs to ensure that the canonical map from the connective K-homology of to the nonconnective K-homology of (both with finite -coefficients) is injective. The condition that the -homology of is concentrated in even degrees implies the injectvity of this map; however, it may hold under more general circumstances (see also Remark LABEL:Gen).
Our strategy makes use of the Davis–Lück unified perspective on the isomorphism conjectures. Some intricate analysis of assembly maps with finite coefficients, which we define in this article (see Definition LABEL:FinAss), is done invoking Suslin’s work on algebraic K-theory [Suslin, SusLoc]. Along the way in Section 1 we produce Künneth type spectral sequences using the machinery of [EKMM] for computing the domains of the assembly maps. Although an extremely trivial case of these spectral sequences is used in this article, we hope that they will be of independent interest. We also observe that for any torsion free Gromov hyperbolic group the canonical algebra homomorphism induces an isomorphism between their algebraic K-theory groups (see Theorem LABEL:mySW). Let us mention that the split -theoretic Novikov conjecture is known to be true in numerous examples. For instance, let be a countable discrete group with a proper left-invariant metric. Thanks to Skandalis–Tu–Yu we know that if admits a uniform embedding into a Hilbert space, then satisfies the split -theoretic Novikov conjecture with coefficients in any separable --algebra [YuCoarseBC, SkaTu]. Guentner–Higson–Weinberger showed that if is a countable discrete subgroup of for any field or of any almost connected Lie group, then satisfies the split -theoretic Novikov conjecture with coefficients in any separable --algebra [GueHigWei]. As a consequence of the above-mentioned reduction principles, we arrive at the following interesting application:
Any countable discrete and torsion free subgroup of a general linear group over an arbitrary field or a similar subgroup of an almost connected Lie group satisfies the split K-theoretic Novikov conjecture over and , i.e., the Loday assembly maps
are split injective. They also satisfy the K-theoretic Novikov conjecture with finite coefficients over and , i.e., the Loday assembly maps and are injective with finite coefficients, provided satisfies the additional hypotheses of Theorem 0.2.
For a countable discrete and torsion free group with a proper left-invariant metric, the above assertions continue to hold if the group admits a uniform embedding into a Hilbert space.
The algebra of Schatten class operators is very important from the viewpoint of higher index theory. The split injectivity result above can also be deduced for many groups (e.g., when is a discrete subgroup of an almost connected Lie group) with arbitrary coefficient algebra from an earlier work of Bartels–Rosenthal [BarRos] (see also some related work of Ji [JiNovikov]). Our result can be regarded as an application of the Baum–Connes conjecture [BaumConnes]; more precisely, that of the (split) injectivity part of the assertion.
Acknowledgements. The author is extremely grateful to G. Yu for his comments on the first draft of this article. The author also wishes to thank P. Baum, R. J. Deeley, R. Meyer, H. Reich, J. M. Rosenberg, A. Valette and C. Westerland for helpful email correspondences. Finally the author is indebted to the anonymous referees for pointing out a mistake in the first draft and the constructive feedback.
1. Some spectral sequences
For any -algebra , there is a symmetric spectrum (in the sense of [HSS]) model of , which is, in addition, a (left) module spectrum over a (commutative) symmetric ring spectrum model of (see Theorem B of [JoaKHom]). Furthermore, there is a unit map from the sphere spectrum to , which is a homomorphism of commutative symmetric ring spectra. After passing to functorial cofibrant replacements (in the -model structures [ShiModel] or the flat stable model structures as in Schwede’s book on symmetric spectra [SchwedeBook]) on the categories of (left) module spectra over the symmetric ring spectra, we may assume that all spectra are cofibrant. Now apply the functorial left Quillen construction, which produces a (cofibrant) -algebra (resp. -module) from a (cofibrant) symmetric ring spectrum (resp. symmetric spectrum) as explained in [SchSModSymSpec]. Thus we obtain a model of as a (left) -module over an -algebra model of , where all -algebras (resp. -modules) are cofibrant. For the details about -algebras and -modules the readers may refer to [EKMM]. Now one may write
using a CW model of . If is a (cofibrant) commutative -algebra and are (cofibrant) -modules, then there is a strongly convergent natural (both in and ) spectral sequence (see Theorem 4.1 of [EKMM])
Here is the resolution degree of and is the internal degree of the graded modules whence it is a right half plane homological spectral sequence.
The symmetric spectra constructed in [JoaKHom] take values in pointed simplicial sets, whereas -modules are spectra valued in based spaces. However, it is known that there is a Quillen equivalence between the category of symmetric spectra valued in pointed simplicial sets and that of symmetric spectra valued in based spaces (see Section 18 of [MMSS]).
Setting , and and observing that we get:
There is a right half plane homological strongly convergent natural spectral sequence
where the degree of is .
There is an identification
which is natural in .
One knows that via the corner embedding . The assertion is now evident from the above spectral sequence (2). ∎
Setting , and in the spectral sequence (1), we get
Apart from the standard Atiyah–Hirzebruch spectral sequences, these Künneth type spectral sequences can potentially be useful for computational purposes for the domain of the Davis–Lück assembly map in certain situations (compare [RosSch1]).
2. Assembly maps with coefficients in topological algebras
We are going to use certain permanence properties to establish the equivalence between the Davis–Lück and the Baum–Connes assembly maps with coefficients in . A key ingredient for us will be the Chabert–Echterhoff–Oyono-Oyono going-down mechanism [CEO]. One obtains very refined permanence properties by localization of triangulated categories following Meyer–Nest [MeyNes], which might be helpful for further generalizations.
Let be a separable and unital -algebra, on which acts trivially. In this case the reduced crossed product simply becomes . In the sequel we denote by . Recall that is defined as a suitable completion of , so that there is a canonical complex algebra homomorphism . Let denote the induced map of K-theory spectra.
By the naturality of the Loday assembly map there is a commutative diagram in the homotopy category of spectra: