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

We use the techniques of Cuntz and Quillen to present a new approach to periodic cyclic homology. Our construction is based on ((Ω•A)[t],d+t⋅ıΔ), a noncommutative equivariant de Rham complex   of an associative algebra AA. Here d is the Karoubi–de Rham differential and ıΔıΔ 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 periodic 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.