Bivariant Chern Character

Definition

The bivariant Chern character is a mathematical construct that generalizes the notion of the Chern character from K-theory to a bivariant setting, allowing for the study of both vector bundles and their morphisms in a coherent way. It connects K-theory with cyclic cohomology, providing a way to compute invariants associated with bundles over spaces and their relationships through continuous maps.

5 Must Know Facts For Your Next Test

1. The bivariant Chern character is defined in terms of two spaces, allowing it to take into account both the source and target vector bundles.
2. It can be viewed as a map from the bivariant K-theory groups to cyclic cohomology groups, thus establishing a deep connection between these mathematical areas.
3. This character plays a crucial role in index theory, particularly in understanding the index of elliptic operators on manifolds.
4. In practical terms, it allows for computations of characteristic classes in situations where traditional methods may not apply due to the presence of morphisms.
5. The bivariant Chern character retains many properties of its univariant counterpart but extends them to encompass more complex relationships between spaces.

Review Questions

• How does the bivariant Chern character extend the concept of the Chern character, and what implications does this have for studying vector bundles?
  • The bivariant Chern character extends the concept of the Chern character by incorporating two spaces, allowing it to analyze morphisms between vector bundles. This extension is significant as it provides tools to compute invariants related to both bundles and maps, making it possible to study more complex interactions in topology. By linking these properties, it enriches our understanding of characteristic classes and their relationships.
• Discuss how the bivariant Chern character is connected to cyclic cohomology and why this connection is important in K-theory.
  • The bivariant Chern character is connected to cyclic cohomology as it serves as a bridge between bivariant K-theory and cohomological invariants. This connection is important because it allows mathematicians to leverage tools from cyclic cohomology for computations within K-theory, facilitating a deeper understanding of geometric and topological structures. Such relationships enhance our ability to compute indices and analyze properties of various spaces and morphisms.
• Evaluate the role of the bivariant Chern character in index theory and its impact on understanding elliptic operators on manifolds.
  • The bivariant Chern character plays a pivotal role in index theory by providing a framework for computing the index of elliptic operators on manifolds. Its ability to connect vector bundles with morphisms means that it can capture information about both differential operators and their corresponding functional spaces. This has significant implications, as it allows mathematicians to derive results regarding solvability and the nature of solutions to partial differential equations on manifolds, ultimately impacting various areas within mathematics and physics.

Related terms

Chern Character: The Chern character is a homomorphism from the K-theory of a topological space to its cohomology ring, representing topological invariants of vector bundles.

KK-Theory: KK-Theory is a bivariant K-theory that provides a framework for studying morphisms between different K-groups, allowing the analysis of spaces and their continuous maps.

Cyclic Cohomology: Cyclic cohomology is an invariant of algebras that captures information about their structure, closely related to K-theory and used in the computation of indices.