Cyclic cohomology

Definition

Cyclic cohomology is a mathematical framework used to study noncommutative algebras, providing a way to compute invariants and establish connections between geometry and topology. This concept links differential forms on noncommutative spaces with the idea of cyclicity, where one can relate cycles and boundaries in a cohomological sense, paving the way for deep results in areas like noncommutative geometry and index theory.

5 Must Know Facts For Your Next Test

1. Cyclic cohomology provides a way to construct invariants that are crucial for understanding the topology of noncommutative algebras.
2. It can be viewed as a generalization of de Rham cohomology to the noncommutative setting, enabling the use of differential forms in this context.
3. The Connes-Chern character is a significant result that connects cyclic cohomology with K-theory, offering a means to compute invariants of noncommutative spaces.
4. Cyclic cohomology plays a vital role in formulating index theorems for noncommutative spaces, bridging concepts from algebra and topology.
5. In commutative settings, cyclic cohomology can recover classical results from topology, showing its foundational importance in linking noncommutative geometry to traditional geometric ideas.

Review Questions

• How does cyclic cohomology relate to differential calculi in the context of noncommutative geometry?
  • Cyclic cohomology extends concepts from differential calculi by allowing the use of differential forms on noncommutative algebras. This connection is significant because it enables one to apply familiar tools from calculus and topology to study spaces that do not behave like traditional manifolds. It effectively captures the essential properties of these algebras while preserving important geometrical features.
• Discuss how cyclic cohomology is utilized within the framework of the Connes-Chern character and its implications in K-theory.
  • The Connes-Chern character provides a bridge between cyclic cohomology and K-theory by establishing a homomorphism that relates cyclic invariants to topological aspects of vector bundles. This relationship has profound implications as it allows mathematicians to compute topological invariants from the perspective of cyclic cohomology, facilitating deeper insights into the structure of noncommutative spaces and their classifications.
• Evaluate the impact of cyclic cohomology on formulating index theorems for noncommutative spaces, considering its role in broader mathematical contexts.
  • Cyclic cohomology significantly impacts index theorems by providing essential tools for understanding operators on noncommutative spaces. This influence extends beyond algebraic structures, linking analytic properties of operators with topological invariants. By enabling mathematicians to compute indices in these settings, cyclic cohomology helps formulate results that unify disparate areas of mathematics, reinforcing its importance in contemporary mathematical research.

Related terms

Chern character:

A homomorphism that associates to each vector bundle a cohomology class, capturing topological information about the bundle and relating it to cyclic cohomology.

Spectral triple: A construction in noncommutative geometry that generalizes the notion of a geometric space, consisting of an algebra, a Hilbert space, and a Dirac operator.

De Rham cohomology: A cohomology theory that computes topological invariants of smooth manifolds using differential forms, providing insight into both classical and noncommutative geometries.