## Illinois Journal of Mathematics

### Mapping the homology of a group to the $K$-theory of its $C\sp *$-algebra

Michel Matthey

#### Abstract

For a CW-complex $X$ and for $0\leq j\leq 2$, we construct natural homomorphisms $\beta_{j}^{X}\colon H_{j}(X;\,\mathbb{Z}) \longrightarrow K_{j}(X)$ that are rationally right-inverses of the Chern character. We show that $\beta_{j}^{X}$ is injective for $j=0$ and $j=1$. The case $j=3$ is treated using $\mathbb{Z}[\frac12]$-coefficients. The study of these maps is motivated by the connection with the Baum-Connes conjecture on the $K$-theory of group $C^{*}$-algebras.

#### Article information

Source
Illinois J. Math., Volume 46, Number 3 (2002), 953-977.

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258130995

Digital Object Identifier
doi:10.1215/ijm/1258130995

Mathematical Reviews number (MathSciNet)
MR1951251

Zentralblatt MATH identifier
1021.19003

#### Citation

Matthey, Michel. Mapping the homology of a group to the $K$-theory of its $C\sp *$-algebra. Illinois J. Math. 46 (2002), no. 3, 953--977. doi:10.1215/ijm/1258130995. https://projecteuclid.org/euclid.ijm/1258130995