Chern Character in Cyclic Cohomology

Definition

The Chern character in cyclic cohomology is a topological invariant that relates the geometry of a vector bundle to cohomological data, particularly in noncommutative geometry. It serves as a crucial tool for bridging the gap between differential geometry and algebraic topology, allowing for the study of the invariants of noncommutative spaces through cyclic cohomology, which generalizes de Rham cohomology for noncommutative algebras.

5 Must Know Facts For Your Next Test

1. The Chern character can be computed using a formula involving traces of the exponential of curvature forms associated with a connection on a vector bundle.
2. In cyclic cohomology, the Chern character provides a way to define invariants that can be associated with noncommutative manifolds, linking their geometric and algebraic properties.
3. It satisfies several important properties, including compatibility with pullbacks and additivity over direct sums of vector bundles.
4. The Chern character is closely related to other invariants, such as the Todd class and the Chern classes, providing deeper insight into the topology of the underlying space.
5. In the context of noncommutative geometry, the Chern character helps define connections between different areas such as index theory and the Atiyah-Singer index theorem.

Review Questions

• How does the Chern character serve as a bridge between differential geometry and cyclic cohomology?
  • The Chern character acts as a bridge by translating geometric properties of vector bundles into cohomological data through cyclic cohomology. It captures essential topological invariants that can be computed using curvature forms, allowing one to study vector bundles on both commutative and noncommutative spaces. This connection enriches our understanding of how classical geometry principles apply in more abstract settings, showing that cyclic cohomology can encapsulate important geometric information.
• Discuss the significance of calculating the Chern character in the context of noncommutative geometry.
  • Calculating the Chern character is significant in noncommutative geometry as it allows us to define topological invariants for noncommutative spaces, which do not have classical geometric interpretations. These invariants help reveal how geometric concepts like curvature and connections manifest in noncommutative settings. The ability to derive results about the topology of noncommutative algebras using tools like cyclic cohomology opens new avenues for research in mathematical physics and operator algebras.
• Evaluate how the properties of the Chern character influence its application in modern mathematical theories.
  • The properties of the Chern character significantly influence its application in modern mathematical theories by providing essential tools for understanding complex relationships between geometry and topology. Its compatibility with pullbacks and additivity over direct sums enables mathematicians to work with complicated bundles systematically. Additionally, its connections with other characteristic classes and invariants enhance its role in index theory, especially through results like the Atiyah-Singer index theorem, which links analysis, topology, and geometry together in profound ways.

Related terms

Cyclic Cohomology: A homology theory for associative algebras that extends the notion of cohomology by incorporating cycles that are invariant under cyclic permutations.

Characteristic Class: A characteristic class is a way of associating a topological invariant to a vector bundle, which captures essential geometric information about the bundle.

Noncommutative Geometry: An area of mathematics that generalizes geometric concepts to spaces where the coordinates do not commute, often modeled by noncommutative algebras.