An Introduction to K-theory and Cyclic Cohomology

TL;DR

This paper provides an accessible introduction to K-theory and cyclic cohomology, covering their basic ideas, examples, properties, and key theorems, with detailed calculations and applications in noncommutative geometry.

Contribution

It offers a comprehensive exposition of K-theory and cyclic cohomology, including new insights into their interrelation and computational techniques, especially in the context of noncommutative algebras.

Findings

Description of the Chern homomorphism and index theorems

Calculation of cyclic and reduced cyclic cohomology

Application of Goodwillie's theorem to semi-direct product algebras

Abstract

These lecture notes contain an exposition of basic ideas of K-theory and cyclic cohomology. I begin with a list of examples of various situations in which the K-functor of Grothendieck appears naturally, including the rudiments of the topological and algebraic K-theory, K-theory of C^*-algebras, and K-homology. I then discuss elementary properties of cyclic cohomology using the Cuntz-Quillen version of the calculus of noncommutative differential forms on an algebra. As an example of the relation between the two theories we describe the Chern homomorphism and various index-theorem type statements. The remainder of the notes contains some more detailed calculations in cyclic and reduced cyclic cohomology. A key tool in this part is Goodwillie's theorem on the cyclic complex of a semi-direct product algebra. The final chapter gives an exposition of the entire cyclic cohomology of Banach algebras from the point of view of supertraces on the Cuntz algebra. The results discussed here include the simplicial normalization of the entire cyclic cohomology, homotopy invariance and the action of derivations.