Annals of K-Theory
- Ann. K-Theory
- Volume 1, Number 4 (2016), 339-377.
Splitting the relative assembly map, Nil-terms and involutions
We show that the relative Farrell–Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for algebraic -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 -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.
Ann. K-Theory, Volume 1, Number 4 (2016), 339-377.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 18F25: Algebraic $K$-theory and L-theory [See also 11Exx, 11R70, 11S70, 12- XX, 13D15, 14Cxx, 16E20, 19-XX, 46L80, 57R65, 57R67] 19A31: $K_0$ of group rings and orders 19B28: $K_1$of group rings and orders [See also 57Q10] 19D35: Negative $K$-theory, NK and Nil
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