A Caveat on the Isomorphism Conjecture in L-theory

Farrell, Tom; Jones, L. E.; Lück, Wolfgang

doi:10.1515/form.2002.019

Cited by 15 publications

(18 citation statements)

References 12 publications

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“…The Isomorphism Conjecture of Farrell and Jones is not true for the decorations j = 0, 1, 2, which correspond to the decorations p, h and s appearing in the literature [13].…”

Section: Review Of Spaces Over a Category And Assembly Mapsmentioning

confidence: 99%

The p-chain Spectral Sequence

Davis¹,

Lueck²

2003

K-Theory

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We introduce a new spectral sequence called the p-chain spectral sequence which converges to the (co-)homology of a contravariant C-space with coefficients in a covariant C-spectrum for a small category C. It is different from the corresponding Atiyah-Hirzebruch type spectral sequence. It can be used in combination with the Isomorphism Conjectures of Baum-Connes and Farrell-Jones to compute algebraic K-and L-groups of group rings and topological K-groups of reduced group C * -algebras.

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“…The Isomorphism Conjecture of Farrell and Jones is not true for the decorations j = 0, 1, 2, which correspond to the decorations p, h and s appearing in the literature [13].…”

Section: Review Of Spaces Over a Category And Assembly Mapsmentioning

confidence: 99%

The p-chain Spectral Sequence

Davis¹,

Lueck²

2003

K-Theory

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“…The original source for (Fibered) Farrell-Jones Conjecture is [28]. The corresponding conjecture is false if one replaces the decoration −∞ with the decoration p, h or s (see [29] ) is strongly continuous. Theorem 2.72.7 and Lemma 8.2 imply that for any ring R and any direct system of groups {G i | i ∈ I} (with not necessarily injective structure maps), G = colim i∈I G i satisfies the Fibered Farrell-Jones Conjecture for algebraic L-theory with coefficients in R if each group G i does.…”

Section: The Farrell-jones Conjecture For L-theorymentioning

confidence: 99%

On the Farrell-Jones Conjecture and its applications

Bartels

Lück

Reich

2007

Journal of Topology

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Abstract. We present the status of the Farrell-Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true and study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture and conjectures generalizing Moody's Induction Theorem. Thus we extend the class of groups for which these conjectures are known considerably. Mathematics Subject Classification (2000). 19Dxx, 19A31,19B28Keywords. Algebraic K-theory of group rings with arbitrary coefficients, Farrell-Jones Conjecture, Bass Conjecture, Kaplansky Conjecture, Moody's Induction Theorem. Introduction and statements of results0.1. Background. The Farrell-Jones Conjecture for algebraic K-theory predicts the structure of K n (RG) for a group G and a ring R. There is also an L-theory version. For applications in topology and geometry the case R = Z is the most important one since many topological invariants of manifolds and CW -complexes such as the finiteness obstruction, the Whitehead torsion and the surgery obstruction take values in the algebraic K-or L-theory of the integral group ring Zπ of the fundamental group π. The Farrell-Jones Conjecture for R = Z implies several famous conjectures, e.g., the Novikov Conjecture, (in high dimensions) the Borel Conjecture, and the triviality of compact h-cobordisms with torsionfree fundamental group. On the other hand proofs of the Farrell-Jones Conjecture for certain groups often rely on working with integral coefficients since they are based on these geometric connections. This is the reason why more is known about the algebraic K-and L-theory of ZG than of CG which is in some sense surprising since CG has better ring theoretic properties than ZG. For the status of the Farrell-Jones Conjecture with coefficient in Z we refer for instance to [45, Sections 5.2 and 5.3].Recently the geometric approaches have been generalized so far that they also apply to other coefficient rings than Z (see for instance , BartelsReich [8], Bartels-Lück-Reich [6], Quinn [54]). This is interesting for algebraic and ring theoretic applications, where one would like to consider for example fields, rings of integers in algebraic number fields and integral domains. The purpose of

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“…There are counterexamples to the Farrell-Jones Conjecture for the other decorations p, h and s (see [16]). …”

Section: Algebraic K -Theorymentioning

confidence: 99%

K– andL–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by ℤ∕4

Lueck

2005

Geom. Topol.

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We compute the group homology, the topological K -theory of the reduced C * -algebra, the algebraic K -theory and the algebraic L-theory of the group ring of the semi-direct product of the three-dimensional discrete Heisenberg group by Z/4. These computations will follow from the more general treatment of a certain class of groups G which occur as extensions 1 → K → G → Q → 1 of a torsionfree group K by a group Q which satisfies certain assumptions. The key ingredients are the Baum-Connes and Farrell-Jones Conjectures and methods from equivariant algebraic topology.

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A Caveat on the Isomorphism Conjecture in L-theory

Cited by 15 publications

References 12 publications

The p-chain Spectral Sequence

The p-chain Spectral Sequence

On the Farrell-Jones Conjecture and its applications

K– andL–theory of the semi-direct product of the discrete 3–dimensional Heisenberg group by ℤ∕4

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