Annals of K-Theory

Splitting the relative assembly map, Nil-terms and involutions

Abstract

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.

Article information

Source
Ann. K-Theory, Volume 1, Number 4 (2016), 339-377.

Dates
Received: 12 January 2015
Revised: 16 September 2015
Accepted: 5 October 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.akt/1510841596

Digital Object Identifier
doi:10.2140/akt.2016.1.339

Mathematical Reviews number (MathSciNet)
MR3536432

Zentralblatt MATH identifier
06617207

Citation

Lück, Wolfgang; Steimle, Wolfgang. Splitting the relative assembly map, Nil-terms and involutions. Ann. K-Theory 1 (2016), no. 4, 339--377. doi:10.2140/akt.2016.1.339. https://projecteuclid.org/euclid.akt/1510841596

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