Algebraic K ‐Theory In Low Degree and The Novikov Assembly Map

We prove that the Novikov assembly map for a group Γ factorizes, in ‘low homological degree’, through the algebraic K ‐theory of its integral group ring. In homological degree 2, this answers a question posed by N. Higson and P. Julg. As a direct application, we prove that if Γ is torsion‐free and satisfies the Baum‐Connes conjecture, then the homology group H 1 (Γ; Z) injects in K 1 ( C r ∗ Γ ) and in K 1 a l g ( A ) , for any ring A such that Z Γ ⊆ A ⊆ C r ∗ Γ . If moreover B Γ is of dimension less than or equal to 4, then we show that H 2 (Γ; Z) injects in K 0 ( C r ∗ Γ ) and in K 2 a l g ( A ) / Δ 2 , where A is as before, and Δ 2 is generated by the Steinberg symbols {γ,γ}, for Γ∈Γ. 2000 Mathematical Subject Classification : primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.