ALGEBRAIC K-THEORY IN LOW DEGREE AND THE NOVIKOV ASSEMBLY MAP

Abstract

We prove that the Novikov assembly map for a group $\Gamma$ 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 $\Gamma$ is torsion-free and satisfies the Baum-Connes conjecture, then the homology group $H_{1}(\Gamma;\,\mathbb{Z})$ injects in $K_{1}(C^{*}_{r}\Gamma)$ and in $K_{1}^{\rm alg}(A)$, for any ring $A$ such that $\mathbb{Z}\Gamma\subseteq A\subseteq C^{*}_{r}\Gamma$. If moreover $B\Gamma$ is of dimension less than or equal to 4, then we show that $H_{2}(\Gamma;\,\mathbb{Z})$ injects in $K_{0}(C^{*}_{r}\Gamma)$ and in $K_{2}^{\rm alg}(A)/\Delta_{2}$, where $A$ is as before, and $\Delta_{2}$ is generated by the Steinberg symbols $\{\gamma,\,\gamma\}$, for $\gamma\in\Gamma$.

2000 Mathematical Subject Classification: primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.