arXiv:1501.02602v1 [math.KT] 12 Jan 2015 SPLITTING THE RELATIVE ASSEMBLY MAP, NIL-TERMS AND INVOLUTIONS WOLFGANG L ¨ UCK AND WOLFGANG STEIMLE Abstract. We show that the relative Farrell-Jones assembly map from the family of finite subgroups to the family of virtually cyclic subgroups for alge- braic 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 co- homology 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. Introduction 0.1. Motivation. The K-theoretic Farrell-Jones Conjecture for a group G and a ring R predicts that the assembly map asmb n : H G n (E G; K R ) → H G n (G/G; K R )= K n (RG) is an isomorphism for all n ∈ Z. Here E G = E VC (G) is the classifying space for the family VC of virtually cyclic subgroups and H G n (-; K G R ) is the G-homology theory associated to a specific covariant functor K G R from the orbit category Or(G) to the category of spectra Spectra. It satisfies H G n (G/H ; K G R )= π n (K G (G/H )) = K n (RH ) for any subgroup H ⊆ G and n ∈ Z. The assembly map is induced by the projection E G → G/G. The original source for the Farrell-Jones Conjecture is the paper by Farrell-Jones [7, 1.6 on page 257 and 1.7 on page 262]. More information about the Farrell-Jones Conjecture and the classifying spaces for families can be found for instance in the survey articles [16] and [18]. Let E G = E Fin (G) be the classifying space for the family F in of finite subgroups, sometimes also called the classifying space for proper G-actions. The up to G- homotopy unique G-map E G → E G induces a so-called relative assembly map asmb n : H G n (E G; K R ) → H G n (E G; K R ). The main result of a paper by Bartels [3, Theorem 1.3] says that asmb n is split injective for all n ∈ Z. In this paper we improve on this result in two different directions: First we generalize from the context of rings R to the context of additive categories A with G-action. This improvement allows to consider twisted group rings and involutions twisted by an orientation homomorphism G → {±1}; moreover one obtains better inheritance properties and gets fibered versions for free. Date : January, 2015. 2000 Mathematics Subject Classification. 18F25,19A31,19B28,19D35. Key words and phrases. splitting relative K-theoretic assemby maps, rational vanishing and Tate cohomology of the relative Nil-term. 1