Abstract:
By the classical Hurewicz theorem, for any homology theory satisfying the Eilenberg-MacLane axioms there is a natural homomorphism from the fundamental group to the first homology group. This talk is devoted to a generalization of this construction. We construct a natural homomorphism from the fundamental group of a $C^*$-algebra to its $K$-homology. In the commutative case this map agrees with the classical Hurewicz homomorphism. There are nontrivial examples of the Hurewicz homomorphism for $C^*$-algebras with non-Hausdorff spectrum. In particular, the Hurewicz homomorphism for the Kronecker foliation algebra is an isomorphism.
