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

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 ...

The Homology of Cyclic Products

On the cyclic Homology of multiplier Hopf algebras

In this paper, we will study the theory of cyclic homology for regular multiplier Hopf algebras. We associate a cyclic module to a triple $(mathcal{R},mathcal{H},mathcal{X})$ consisting of a regular multiplier Hopf algebra $mathcal{H}$, a left $mathcal{H}$-comodule algebra $mathcal{R}$, and a unital left $mathcal{H}$-module $mathcal{X}$ which is also a unital algebra. First, we construct a para...

Cyclic homology and equivariant homology

The purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of Connes [9-11], see also Loday and Quillen [20], and "IF equivariant homology and cohomology theories. Here II" is the circle group. The most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precis...

CYCLIC COMODULES, THE HOMOLOGY OF j, AND j-HOMOLOGY

This short paper arose as a warm-up to the more difficult computations at the primes 2 and 3 of connective versions of Behrens’ Q(2) spectrum, a spectrum conjecturally describing half of the K(2)-local sphere [2]. The natural starting case is to understand the K(1)-local story, giving rise to the present discussion of the homology of the connective image of J spectrum. The homology results are ...