There is a great amount of work on the cyclic homology of crossed-product algebras, by Baum-Connes, Brylinski-Nistor, Connes, Crainic, Elliott-Natsume-Nest, Feigin-Tsygan, Getzler-Jones, Nest, Nistor, among others. However, at the exception of the characteristic map of Connes from the early 80s, we don’t have explicit chain maps that produce isomorphisms at the level of homology and provide us with geometric constructions of cyclic cycles in the case of group actions on manifolds or varieties.

The aim of this talk is to present the construction of explicit quasi-isomorphisms for crossed products associated with actions of discrete groups. Along the way we recover and clarify various earlier results (in the sense that we obtain explicit chain maps that yield quasi-isomorphisms). In particular, we recover the spectral sequences of Feigin-Tsygan and Getzler-Jones, and derive an additional spectral sequence.

In the case of group actions on manifolds we have an explicit description of cyclic homology and periodic cyclic homology. In the finite order case, the results are expressed in terms of what we call ``mixed equivariant homology”, which interpolates group homology and de Rham cohomology. This is actually the natural receptacle for a cap product of group homology with equivariant cohomology. As a result taking cap products of group cycles with equivariant characteristic classes naturally gives to a geometric construction of cyclic cycles. For the periodic cyclic homology we recover earlier results of Connes and Brylinski-Nistor via a Poincar\'e duality argument. For the non-periodic cyclic homology the results seem to be new. In the infinite order case, we fix and simplify the misidentification of cyclic homology by Crainic.