Free products, cyclic homology, and the Gauss-Manin connection

TL;DR

This paper introduces a novel approach to cyclic homology using a noncommutative equivariant de Rham complex, simplifying the construction of the Gauss-Manin connection and exploring free-product deformations of associative algebras.

Contribution

It presents a new cyclic homology framework avoiding the Connes differential and introduces free-product deformations over noncommutative bases.

Findings

A new cyclic homology approach based on a noncommutative de Rham complex.

Explicit construction of the Gauss-Manin connection for associative algebra families.

Introduction of free-product deformations with examples from preprojective and surface group algebras.

Abstract

We present a new approach to cyclic homology that does not involve the Connes differential and is based on a `noncommutative equivariant de Rham complex' of an associative algebra. The differential in that complex is a sum of the Karoubi-de Rham differential, which replaces the Connes differential, and another operation analogous to contraction with a vector field. As a byproduct, we give a simple explicit construction of the Gauss-Manin connection, introduced earlier by E. Getzler, on the relative cyclic homology of a flat family of associative algebras over a central base ring. We introduce and study `free-product deformations' of an associative algebra, a new type of deformation over a not necessarily commutative base ring. Natural examples of free-product deformations arise from preprojective algebras and group algebras for compact surface groups.