Abstract. In this paper we study continuous bundles of C*-algebras which are noncommutative analogues of principal torus bundles. We show that all such bundles, although in general being very far away from being locally trivial bundles, are at least locally RKK-trivial. Using earlier results of Echterhoff and Williams, we shall give a complete classification of principal non-commutative torus bundles up to T n -equivariant Morita equivalence. We then study these bundles as topological fibrations (forgetting the group action) and give necessary and sufficient conditions for any non-commutative principal torus bundle being RKK-equivalent to a commutative one. As an application of our methods we shall also give a K-theoretic characterization of those principal T n -bundles with H-flux, as studied by Mathai and Rosenberg which possess "classical" T -duals.

We study the connection between the Baum-Connes conjecture for a locally compact group G with coefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product A r G. The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.

Let G be a compact Lie-group, X a compact G-CW-complex. We define equivariant geometric K-homology groups K G * (X), using an obvious equivariant version of the (M, E, f )-picture of Baum-Douglas for K-homology. We define explicit natural transformations to and from equivariant K-homology defined via KK-theory (the "official" equivariant K-homology groups) and show that these are isomorphisms.Keywords Equivariant K-homology · Geometric K-homology · G-CW-complex · Equivariant Baum-Douglas cycles · Equivariant (M, E, f ) · Gysin homomorphism in equivariant K-theory · Poincaré duality in equivariant K-theory Mathematics Subject Classification (2000)19K33 · 19K35 · 19K56 · 19L47 · 55N20 · 55R91 Communicated by B. Richter.