# Cyclic Homology and Group Actions

## TL;DR

This paper constructs explicit quasi-isomorphisms to compute cyclic homology of crossed-product algebras from group actions, extending to various algebraic and geometric contexts, and introduces new homological tools and spectral sequences.

## Contribution

It introduces the concept of triangular S-modules and extends cyclic homology computations to actions on manifolds and varieties, with explicit constructions and new spectral sequences.

## Key findings

- Explicit quasi-isomorphisms for cyclic homology of crossed-products
- Introduction of triangular S-modules as a new homological tool
- Development of new spectral sequences for cyclic homology

## Abstract

In this paper we present the construction of explicit quasi-isomorphisms that compute the cyclic homology and periodic cyclic homology of crossed-product algebras associated with (discrete) group actions. In the first part we deal with algebraic crossed-products associated with group actions on unital algebras over any ring $k\supset \mathbb{Q}$. In the second part, we extend the results to actions on locally convex algebras. We then deal with crossed-products associated with group actions on manifolds and smooth varieties. For the finite order components, the results are expressed in terms of what we call "mixed equivariant cohomology". This "mixed" theory mediates between group homology and de Rham cohomology. It is naturally related to equivariant cohomology, and so we obtain explicit constructions of cyclic cycles out of equivariant characteristic classes. For the infinite order components, we simplify and correct the misidentification of Crainic. An important new homological tool is the notion of "triangular $S$-module". This is a natural generalization of the cylindrical complexes of Getzler-Jones. It combines the mixed complexes of Burghelea-Kassel and parachain complexes of Getzler-Jones with the $S$-modules of Kassel-Jones. There are spectral sequences naturally associated with triangular $S$-modules. In particular, this allows us to recover spectral sequences Feigin-Tsygan and Getzler-Jones and leads us to a new spectral sequence.