Annals of K-Theory

Splitting the relative assembly map, Nil-terms and involutions

Wolfgang Lück and Wolfgang Steimle

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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 [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

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

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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

splitting relative $K$-theoretic assembly maps rational vanishing and Tate cohomology of the relative Nil-term


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.

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