Chern characters for proper equivariant homology theories and applications to K- and L-theory

Gruyter, Walter de; Lück, Wolfgang

doi:10.1515/crll.2002.015

Cited by 67 publications

(124 citation statements)

References 22 publications

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“…In particular, the cokernel of (0.2) contains the cokernel of (0.3) in the situation of Theorem A. This cokernel can be evaluated using an Atiyah-Hirzebruch spectral sequence (see [DL98,Theorem 4.7]), or computed rationally using the equivariant Chern character from [Lüc02,Theorem 0.3], [LR05,Theorem 173]. If R = Z and n = 1, then the cokernel of (0.2) is the Whitehead group.…”

Section: Introductionmentioning

confidence: 99%

On the K-theory of groups with finite asymptotic dimension

Bartels

Rosenthal

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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Section: Introductionmentioning

confidence: 99%

On the K-theory of groups with finite asymptotic dimension

Bartels

Rosenthal

2007

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…In the previous section we have verified that the assumptions of Theorem 0.1 and of Theorem 0.2 in [20] are satisfied in the case where the equivariant homology theory H ?…”

Section: Evaluating the Equivariant Chern Charactermentioning

confidence: 55%

“…We do not know any non-trivial example where the isomorphism statement in Addendum 1.8 applies, i.e. where F contains all (finite and infinite) cyclic groups and where, at the same time, E F (G) has a cocompact model.The second main ingredient of our investigation is the rational computation of equivariant homology theories from [20]. For varying G, our G-homology theories like…”

mentioning

confidence: 99%

“…In Section 6, we review these notions and explain some general principles which allow us to verify 8 WOLFGANG LÜCK AND HOLGER REICH that G-homology theories like the ones we are interested in indeed admit induction and Mackey structures. In particular, Theorems 0.1 and 0.2 in [20] apply and yield an explicit computation of H G * (EG)⊗ Z Q. In Section 7, we review this computation and discuss a simplification which occurs in the case of K-theory, Hochschild, cyclic, periodic cyclic and negative cyclic homology, due to the fact that in all these special cases, we have additionally a module structure over the Swan ring.…”

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confidence: 99%

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Detecting $K$-Theory by Cyclic Homology

Lück

Reich

2006

Proc. Lond. Math. Soc.

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Abstract. We discuss which part of the rationalized algebraic K-theory of a group ring is detected via trace maps to Hochschild homology, cyclic homology, periodic cyclic or negative cyclic homology. Key words: algebraic K-theory of group rings, Hochschild homology, cyclic homology, trace maps. Mathematics Subject Classification 2000: 19D55. Dedicated to memory of Michel Matthey. Introduction and statement of resultsFix a commutative ring k, referred to as the ground ring. Let R be a k-algebra, i.e. an associative ring R together with a unital ring homomorphism from k to the center of R. We denote by HH ⊗ k * (R) the Hochschild homology of R relative to the ground ring k, and similarly by HC ⊗ k * (R), HP ⊗ k * (R) and HN ⊗ k * (R) the cyclic, the periodic cyclic and the negative cyclic homology of R relative to k. Hochschild homology receives a map from the algebraic K-theory, which is known as the Dennis trace map. There are variants of the Dennis trace taking values in cyclic, periodic cyclic and negative cyclic homology (sometimes called Chern characters), as displayed in the following commutative diagram. For the definition of these maps, see [18, Chapters 8 and 11] and Section 5 below. In the following we will focus on the case of group rings RG, where G is a group and we refer to the k-algebra R as the coefficient ring. We investigate the following question.Question 0.2. Which part of K * (RG) ⊗ Z Q can be detected using linear trace invariants like the Dennis trace to Hochschild homology, or its variants with values in cyclic homology, periodic cyclic homology and negative cyclic homology ?For any group G, we prove "detection results", which state that certain parts of K * (RG) ⊗ Z Q can be detected by the trace maps in diagram 0.1, accompanied by "vanishing results", which state that a complement of the part which is then known to be detected is mapped to zero. For the detection results, we only make assumptions on the coefficient ring R, whereas for the vanishing results we additionally need the Farrell-Jones Conjecture for RG as an input, compare Example 1.2. Modulo the Farrell-Jones Conjecture, we will give a complete answer to Question 0.2 for instance in the case of Hochschild and cyclic homology, when the coefficient ring R is an algebraic number field F or its ring of integers O F . We will also give partial results for periodic cyclic and negative cyclic homology. All detection results are obtained by using only the Dennis trace with values in HH ⊗ k * (RG), whereas all vanishing results hold even for the trace with values in HN ⊗ Z * (RG), which, in view of diagram (0.1), can be viewed as the best among the considered trace invariants. (Note that for a k-algebra R every homomorphism k ′ → k of commutative rings leads to a homomorphism HN ⊗ k ′ * (R) → HN ⊗ k * (R). Similar for Hochschild, cyclic and periodic cyclic homology.) We have no example where the extra effort that goes into the construction of the variants with values in cyclic, periodic cyclic or negative cyclic homology yields mo...

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“…Hence, the source of the assembly map appearing in the FarrellJones conjecture can be computed in two steps, involving the computation of H G n (EG; K R ) and the computation of the remaining term which we denote by H G n (EG → EG; K R ). Rationally, H G n (EG; K R ) can be computed using equivariant Chern characters (see [14,Section 1]) or with the help of nice models for EG.…”

Section: Models For E Sf G (G)mentioning

confidence: 99%

On The Classifying Space of The Family of Virtually Cyclic Subgroups

Lück¹,

Weiermann²

2012

Pure and Applied Mathematics Quarterly

134

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We study the minimal dimension of the classifying space of the family of virtually cyclic subgroups of a discrete group. We give a complete answer for instance if the group is virtually poly-Z, word-hyperbolic or countable locally virtually cyclic. We give examples of groups for which the difference of the minimal dimensions of the classifying spaces of virtually cyclic and of finite subgroups is −1, 0 and 1, and show in many cases that no other values can occur.Key words: classifying spaces of families, dimensions, virtually cyclic subgroups Mathematics Subject Classification 2000: 55R35, 57S99, 20F65, 18G99. IntroductionIn this paper, we study the classifying spaces EG = E VCY (G) of the family of virtually cyclic subgroups of a group G. We are mainly interested in the minimal dimension hdim G (EG) that a G-CW -model for EG can have. The classifying space for proper G-actions EG = E F IN (G) has already been studied intensively in the literature. The spaces EG and EG appear in the source of the assembly maps whose bijectivity is predicted by the Baum-Connes conjecture and the Farrell-Jones conjecture, respectively. Hence the analysis of EG and EG is important for computations of K-and L-groups of reduced group C * -algebras and of group rings (see Section 6). These spaces can also be viewed as invariants * email: lueck@math.uni-muenster.de, michi@uni-muenster.de www: http://www.math.uni-muenster.de/u/lueck/ FAX: 49 251 8338370 1 of G, and one would like to understand what properties of G are reflected in the geometry and homotopy theoretic properties of EG and EG.The main results of this manuscript aim at the computation of the homotopy dimension hdim G (EG) (see Definition 4.1), which is much more difficult than in the case of hdim G (EG). It has lead to some (at least for us) surprising phenomenons, already for some nice groups which can be described explicitly and have interesting geometry.After briefly recalling the notion of EG and EG in Section 1, we investigate in Section 2 how one can build E G (G) out of E F (G) for families F ⊆ G. We construct a G-pushoutfor a certain set M of maximal virtually cyclic subgroups of G, provided that G satisfies the condition (M F IN ⊆VCY ) that every infinite virtually cyclic subgroup H is contained in a unique maximal infinite virtually cyclic subgroup H max . For a subgroup H ⊆ G, we denote by N G H := {g ∈ G | gHg −1 = H} its normalizer and put W G H := N G H/H. We will say that G satisfies (NM F IN ⊆VCY ) if it satisfies (M F IN ⊆VCY ) and N G V = V , i.e., W G V = {1}, holds for all V ∈ M. In Section 3, we give a criterion for G to satisfy (NM F IN ⊆VCY ).Section 4 is devoted to the construction of models for EG and EG from models for EG i and EG i , respectively, if G is the directed union of the directed system {G i | i ∈ I} of subgroups.In Section 5, we prove various results concerning hdim G (EG) and give some examples. In Subsection 5.1, we proveWhile one knows that hdimand give examples showing that this inequality cannot be improved in general. In S...

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Chern characters for proper equivariant homology theories and applications to K- and L-theory

Cited by 67 publications

References 22 publications

On the K-theory of groups with finite asymptotic dimension

On the K-theory of groups with finite asymptotic dimension

Detecting $K$-Theory by Cyclic Homology

On The Classifying Space of The Family of Virtually Cyclic Subgroups

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