Free Products, Cyclic Homology, and the Gauss-manin Connection

We present a new approach to cyclic homology that does not involve Connes’ differential and is based on (Ω q A)[u], d + u · ı∆, a noncommutative equivariant de Rham complex of an associative algebra A. Here d is the Karoubi-de Rham differential, which replaces the Connes differential, and ı∆ is an 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.