## Illinois Journal of Mathematics

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

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

#### 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

**Permanent link to this document**

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

**Mathematical Reviews number (MathSciNet)**

MR1951251

**Zentralblatt MATH identifier**

1021.19003

**Subjects**

Primary: 19K56: Index theory [See also 58J20, 58J22]

Secondary: 19L10: Riemann-Roch theorems, Chern characters 19L41: Connective $K$-theory, cobordism [See also 55N22] 55S45: Postnikov systems, $k$-invariants 57R20: Characteristic classes and numbers

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