Free products, cyclic homology, and the Gauss-Manin connection
arXiv:0803.3655
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.
Some corrections made and additional remarks added. This paper supercedes an earlier paper math.KT/0505236 by one of the authors. Includes an appendix by Boris Tsygan in v4. Final version, to appear in Advances in Math