The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory

Lück, Wolfgang; Reich, Holger

doi:10.1007/978-3-540-27855-9_15

Cited by 159 publications

(180 citation statements)

References 282 publications

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Section: Theorem 015 (Vanishing Results For Integral Coefficients)mentioning

confidence: 99%

“…Before we 6 WOLFGANG LÜCK AND HOLGER REICH explain the general strategy behind our results we briefly explain the concept of a generalized assembly map; for more details the reader is referred to [8] and [22, Section 2 and 6].…”

Section: Outline Of the Methodsmentioning

confidence: 99%

“…We remark that rationally, the Farrell-Jones Conjecture for K * (ZG) is known in many cases, for example for every subgroup G of a discrete cocompact subgroup of a virtually connected Lie group [13]. For a survey of known results about the Farrell-Jones Conjecture, we refer the reader to [22].Remark 0.16. There are further trace invariants (or Chern characters) given by maps ch n,r : K n (RG) → HC ⊗ k n+2r (RG), for fixed n, r ≥ 0, see [18, 8.4.6 on page 272 and 11.4.3 on page 371].…”

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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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Section: Theorem 015 (Vanishing Results For Integral Coefficients)mentioning

confidence: 99%

Section: Outline Of the Methodsmentioning

confidence: 99%

mentioning

confidence: 99%

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Detecting $K$-Theory by Cyclic Homology

Lück

Reich

2006

Proc. Lond. Math. Soc.

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“…is an isomorphism for any group G, any ring R and F the family of virtually cyclic subgroups of G [4]. …”

Section: Remarkmentioning

confidence: 99%

Note on the injectivity of the Loday assembly map

Ullmann

2017

Journal of Algebra

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“…This includes the vanishing of the Whitehead Group for torsion-free groups and the Borel Conjecture about the rigidity of aspherical manifolds. We refer to [LR05,BLR08b] for details. Therefore, it is interesting to know the Farrell-Jones Conjecture for as many rings R, called the coefficients, and groups G as possible.…”

Section: Controlled Algebra For Discrete Ringsmentioning

confidence: 99%

Controlled algebra for simplicial rings and algebraic K-theory

Ullmann¹

Geometric and Cohomological Group Theory

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We develop a version of controlled algebra for simplicial rings. This generalizes the methods which lead to successful proofs of the algebraic K-theory isomorphism conjecture (Farrell-Jones Conjecture) for a large class of groups. This is the first step to prove the algebraic K-theory isomorphism conjecture for simplicial rings. We construct a category of controlled simplicial modules, show that it has the structure of a Waldhausen category and discuss its algebraic K-theory.We lay emphasis on detailed proofs. Highlights include the discussion of a simplicial cylinder functor, the gluing lemma, a simplicial mapping telescope to split coherent homotopy idempotents, and a direct proof that a weak equivalence of simplicial rings induces an equivalence on their algebraic K-theory. Because we need a certain cofinality theorem for algebraic K-theory, we provide a proof and show that a certain assumption, sometimes omitted in the literature, is necessary. Last, we remark how our setup relates to ring spectra.

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The Baum–Connes and the Farrell–Jones Conjectures in K- and L-Theory

Abstract: We give a survey of the meaning, status and applications of the Baum-Connes Conjecture about the topological K-theory of the reduced group C * -algebra and the Farrell-Jones Conjecture about the algebraic K-and L-theory of the group ring of a (discrete) group G.

Cited by 159 publications

References 282 publications

Detecting $K$-Theory by Cyclic Homology

Detecting $K$-Theory by Cyclic Homology

Note on the injectivity of the Loday assembly map

Controlled algebra for simplicial rings and algebraic K-theory

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