A twisted Bass–Heller–Swan decomposition for the algebraic <i>K</i>-theory of additive categories

Lück, Wolfgang; Steimle, Wolfgang

doi:10.1515/forum-2013-0146

Cited by 10 publications

(15 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof of the following result can be found in [19]. The case where the Zaction on B is trivial and one considers only K-groups in dimensions n ≤ 1 has already been treated by Ranicki [23, Chapter 10 and 11].…”

Section: The Twisted Bass-heller-swan Theorem For Additive Categoriesmentioning

confidence: 99%

Splitting the relative assembly map, Nil-terms and involutions

Lück

Steimle

2016

AKT

Self Cite

View full text Add to dashboard Cite

We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic K-theory is split injective in the setting where the coefficients are additive categories with group action. This generalizes a result of Bartels for rings as coefficients. We give an explicit description of the relative term. This enables us to show that it vanishes rationally if we take coefficients in a regular ring. Moreover, it is, considered as a Z[Z/2]-module by the involution coming from taking dual modules, an extended module and in particular all its Tate cohomology groups vanish, provided that the infinite virtually cyclic subgroups of type I of G are orientable. The latter condition is for instance satisfied for torsionfree hyperbolic groups.Comment: 30 pages, to appear in Annals of K-theor

show abstract

Section: The Twisted Bass-heller-swan Theorem For Additive Categoriesmentioning

confidence: 99%

Splitting the relative assembly map, Nil-terms and involutions

Lück

Steimle

2016

AKT

Self Cite

View full text Add to dashboard Cite

show abstract

“…This is well-known when the coefficient is a ring and has been recently generalized by Lück-Steimle [26, Theorem 0.1, Remark 0.2] to with coefficient in any additive category. Using this sequence and the assumption that the K-theoretic FJC holds for G with coefficient in R, we see that Denote the Nil-groups N K n (O G (E(G ⋊ β Z), R); β) as defined in [26] by N il Because G is torsion free and satisfies the L-theoretic FJC with coefficient in Z, Theorem B applies and the L-theoretic FJC with coefficient in Z holds for G ⋊ Z. Therefore, by a standard surgery long exact sequence argument, we see that the simple Borel conjecture holds for G ⋊ Z.…”

Section: 2mentioning

confidence: 99%

Topological rigidity for FJ by the infinite cyclic group

Wang

2018

J. Topol. Anal.

View full text Add to dashboard Cite

Abstract. We call a group FJ if it satisfies the K-and L-theoretic Farrell-Jones conjecture with coefficients in Z. We show that if G is FJ, then the simple Borel conjecture (in dimensions ≥ 5) holds for every group of the form G ⋊ Z. If in addition W h(G × Z) = 0, which is true for all known torsion free FJ groups, then the bordism Borel conjecture (in dimensions n ≥ 5) holds for G ⋊ Z. One of the key ingredients in proving these rigidity results is another main result, which says that if a torsion free group G satisfies the L-theoretic Farrell-Jones conjecture with coefficients in Z, then any semi-direct product G⋊Z also satisfies the L-theoretic Farrell-Jones conjecture with coefficients in Z. Our result is indeed more general and implies the L-theoretic Farrell-Jones conjecture with coefficients in additive categories is closed under extensions of torsion free groups. This enables us to extend the class of groups which satisfy the Novikov conjecture.

show abstract

“…In [22,Section 8] it is shown that if A is idempotent complete, then K(χ) is part of a homotopy fiber sequence (6.8)…”

Section: Compatibility With Colimitsmentioning

confidence: 99%

“…It is conceivable that the twisted Bass-Heller-Swan decomposition for connective K-theory, which is described in Theorem 6.1 and whose proof is given in [22], can be extended directly to the non-connective setting described in Theorem 6.2 using Schlichting's non-connective version of K-theory for exact categories and his localization theorem instead of Waldhausen's approximation and fibration theorems.…”

Section: Compatibility With Colimitsmentioning

confidence: 99%

“…We will use this construction to explain how the twisted Bass-Heller-Swan decomposition for connective K-theory of additive categories can be extended to the non-connective version, compare [22]. We will also discuss that the compatibility of the connective K-theory spectrum with filtered colimits passes to the non-connective K-theory spectrum.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Non-connective 𝐾- and Nil-spectra of additive categories

Lück

Steimle

2014

Contemporary Mathematics

Self Cite

View full text Add to dashboard Cite

We present an elementary construction of the non-connective algebraic K-theory spectrum associated to an additive category following the contracted functor approach due to Bass. It comes with a universal property that easily allows us to identify it with other constructions, for instance with the one of Pedersen-Weibel in terms of Z i -graded objects and bounded homomorphisms.2010 Mathematics Subject Classification. 19D35. Key words and phrases. Non-connected algebraic K-theory spectrum of additive categories, Nil-spectra.As J takes values in functors satisfying Condition 1.1, the maps A(j) : E(j) ∧ (S 1 ) + → ZE(j) are natural in j ∈ J . In this way LE(j) also becomes a functor in j, and further applications of Lemma 5.2 show that the induced mapis a weak equivalence. We obtain a commutative diagramwith weak homotopy equivalences as vertical arrows.Iterating this construction leads to a commutative diagram with weak homotopy equivalences as vertical arrows.

show abstract

A twisted Bass–Heller–Swan decomposition for the algebraic K-theory of additive categories

Abstract: We prove a twisted Bass-Heller-Swan decomposition for both the connective and the non-connective -theory spectrum of additive categories.

Cited by 10 publications

References 16 publications

Splitting the relative assembly map, Nil-terms and involutions

Splitting the relative assembly map, Nil-terms and involutions

Topological rigidity for FJ by the infinite cyclic group

Non-connective 𝐾- and Nil-spectra of additive categories

Contact Info

Product

Resources

About