2013/05/08 by Guillermo Cortiñas, Cortiñas, Guillermo, Gisela Tartaglia +1
Mathematics · #FOS: Mathematics #K-Theory and Homology (math.KT) #math.KT
paper · pdf · doi:10.48550/arxiv.1305.1771
13 pages
arxiv created 2013/06/20 · arxiv updated 2013/06/21
Let G be a group, Fin the family of its finite subgroups, and E(G,Fin) the classifying space. Let L1 be the algebra of trace-class operators in an infinite dimensional, separable Hilbert space over the complex numbers. Consider the rational assembly map in homotopy algebraic K-theory HpG(E(G,Fin),KH(L1))⊗\Q → KHp(L1[G])⊗\Q. The rational KH-isomorphism conjecture predicts that the map above is an isomorphism; it follows from a theorem of Yu (see arXiv:1106.3796, arXiv:1202.4999) that it is always injective. In the current article we prove the following. Theorem: Assume that the map above is surjective. Let n≡ p+1\mod 2. Then: i) The assembly map for the trivial family HnG(E(G,1),K(\Z)) → Kn(\Z[G]) is rationally injective. ii) For every number field F, the assembly map HnG(E(G,Fin),K(F)) → Kn(F[G]) is rationally injective. We remark that the K-theory Novikov conjecture asserts that part i) of the theorem above holds for all G, and that part ii) is equivalent to the rational injectivity part of the K-theory Farrell-Jones conjecture for number fields. The idea of the proof of the Theorem is to use an algebraic, equivariant version of Karoubi's multiplicative K-theory, which we introduce in this article.