Ullmann, M.

Wu, X.

English

MPG.PuRe

arXiv:1612.08625v1  [math.KT]  18 Nov 2016NOTE ON THE INJECTIVITY OF THE LODAY ASSEMBLY MAPMARK ULLMANN AND XIAOLEI WUAbstract. We show that for finite groups the Loday assembly map with coefficients infinite fields is in general not injective.Given a group G, a ring R, one can consider the following Loday assembly map [5,Proposition 4.1.1]:α : H∗(BG, K(R)) −→ K∗(R[G])In general, the map is far from isomorphism. For example, the nonvanishing of white-head torsion will imply the assembly is not surjective. But when the group is torsion freeand the ring is regular, it is implied by the Farrell–Jones Conjecture that this is an isomor-phism, see [4] for definitions, and in particular Corollary 67 for the isomorphism result.One might ask, would the map be injective for groups with torsion and regular rings, seefor example [6, Question 1.1]. We show this is not the case even if the group is finite andR is a field.To construct the counterexample, we need Quillen’s calculation for algebraic K-theoryof finite fields.Theorem 1 ([7, Theorem 8]). Let Fq be a finite field with q elements, then for n > 0,Kn(Fq) = 0 when n is even, and Kn(Fq) is a cyclic group of order qi − 1 when n = 2i − 1.Note that since Fq is a field, Kn(Fq) = 0 when n < 0 and K0(Fq) = Z.Theorem 2. Let G be a finite group such that H2(G,Z) is nontrivial and F a finite fieldwith characteristic p which does not divide the order of G, then the assembly map α isnot injective. For example, we can take G = Z/2 × Z/2 and F to be any finite field withcharacteristic p > 2.Proof Since the characteristic of F does not divide the order of G, by Maschke’s theo-rem, FG is semisimple. Since it is finite, by Artin–Wedderburn’s Theorem, FG is a directproduct of matrix rings over finite fields of characteristic p. Note that for any matrix ringMn(E) over a field E, K∗(Mn(E)) = K∗(E). Thus by Theorem 1, we in particular haveKn(FG) = 0 when n is even and K0(FG) = Zd, where d is the number of components in theArtin–Wedderburn Theorem for FG. Note also that Kn(FG) = 0 when n < 0.Now applying Atiyah–Hirzebruch spectral sequence to the left hand side of the assem-bly map H∗(BG, K(F)), E2p,q = Hp(BG, Kq(F)). (abbreviate H∗(G) := H∗(G,Z))12 MARK ULLMANN AND XIAOLEI WUq3 K3(F) ∗ ∗ ∗2 0 0 0 01 K1(F) ∗ ∗ ∗0 K0(F) H1(G) H2(G) H3(G)0 1 2 3 pNote that the assembly map for K0 and K1 is indeed injective and the terms E20,0, E21,0, E20,1survive [4, Lemma 2]. Thus H2(BG, K0(F)) = H2(G) survives in the spectral sequence. Butsince H2(G) , 0 and K2(FG) = 0, the assembly map can not be injective. Remark 3. It would be interesting if one could find groups G such that the assembly mapis not injective for the ring Z. However it seems that there is not much calculation knownin these cases.Remark 4. In general, the Farrell–Jones Conjecture predicts that the assembly mapHGq (EFG, KR) → Kq(R[G])is an isomorphism for any group G, any ring R and F the family of virtually cyclic sub-groups of G [4]. There are results that the family can be made smaller, see [1, Section 7]and [2]. The Transitivity Principle [4, Theorem 65] and the inheritance under subgroupsimplies, that if the Farrell–Jones Conjecture holds for a family F , it holds for all familiescontaining F . There are also injectivity results for smaller families, for example for thefamily of finite groups, cf. [3]. Thus one may think, that the relative assembly map to alarger family always gives an injective map. Our example shows, that this is not true forthe trivial family to the family of all (or finite) subgroups.Acknowledgements. The first author was supported by the DFG–project C1 of CRC647 in Berlin. The second author is supported by the Max Planck Institute for Mathematicsat Bonn.References[1] A. Bartels, F. T. Farrell, and W. Lu¨ck. The Farrell-Jones conjecture for cocompact lattices invirtually connected Lie groups. J. Amer. Math. Soc., 27(2):339–388, 2014.[2] J. F. Davis, F. Quinn and H. Reich. Algebraic K-theory over the infinite dihedral group: a con-trolled topology approach, J. Topology, 4 (2011) 505-528.NOTE ON THE INJECTIVITY OF THE LODAY ASSEMBLY MAP 3[3] D. Kasprowski. On the K-theory of groups with finite decomposition complexity. Proc. Lond.Math. Soc. (3), 110(3):565–592, 2015.[4] W. Lu¨ck and H. Reich. The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory.In Handbook of K-theory. Vol. 1, 2, pages 703–842. Springer, Berlin, 2005.[5] J.-L. Loday, K-the´orie alge´brique et repre´sentations de groupes, Ann. Sci. ´Ecole Norm. Sup. (4)9 (1976), no. 3, 309-377.[6] C. Malkiewich. Coassembly and the K-theory of finite groups. arXiv:1503.06504[7] D. Quillen. On the cohomology and K-theory of the general linear groups over a finite field. Ann.of Math. (2) 96 (1972), 552-586.Freie Universita¨t Berlin, Institut fu¨rMathematik, Arnimallee 7, 14195 Berlin, GermanyE-mail address: mark.ullmann@math.fu-berlin.deMax Planck Institute forMathematics, Vivatsgasse 7, 53111 Bonn, GermanyE-mail address: hsiaolei.wu@mpim-bonn.mpg.de

Note on the injectivity of the Loday assembly map

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