Conner-Floyd Chern character

Definition

The Conner-Floyd Chern character is a homomorphism from the K-theory of a space to the rational cohomology of that space, which generalizes the classical Chern character from vector bundles to topological spaces. This character allows one to express topological invariants in a way that connects K-theory with differential geometry, playing a crucial role in various computations and applications related to K-groups.

5 Must Know Facts For Your Next Test

1. The Conner-Floyd Chern character is defined using the index theory, connecting K-theory with characteristic classes and providing insights into topological invariants.
2. This character can be computed explicitly for specific classes of spaces and vector bundles, which aids in understanding their properties in relation to K-groups.
3. One important property of the Conner-Floyd Chern character is that it respects certain operations on K-theory, such as direct sums and tensor products.
4. The character is particularly useful in the context of smooth manifolds, where it facilitates the computation of K-groups via differential forms and other geometric tools.
5. The Conner-Floyd Chern character has applications in areas like algebraic geometry and mathematical physics, where it helps bridge the gap between topology and geometry.

Review Questions

• How does the Conner-Floyd Chern character extend the concept of the classical Chern character in the context of topological spaces?
  • The Conner-Floyd Chern character extends the classical Chern character by generalizing it from vector bundles to arbitrary topological spaces. While the classical Chern character focuses on characteristic classes associated with vector bundles, the Conner-Floyd version provides a framework for relating these ideas within K-theory, enabling a more comprehensive understanding of topological invariants across diverse spaces. This extension is crucial for establishing connections between algebraic topology and differential geometry.
• Discuss how the Conner-Floyd Chern character can be used to compute K-groups for specific types of spaces or bundles.
  • The Conner-Floyd Chern character can be applied to compute K-groups by translating problems in K-theory into questions about cohomology. For instance, when dealing with smooth manifolds or complex projective varieties, one can use differential forms or characteristic classes to calculate the Conner-Floyd Chern character explicitly. This computation allows for determining relations among various K-groups by linking them to the cohomological properties of these spaces through this homomorphism.
• Evaluate the significance of the Conner-Floyd Chern character in bridging different areas of mathematics, such as algebraic geometry and mathematical physics.
  • The significance of the Conner-Floyd Chern character lies in its ability to connect seemingly disparate fields like algebraic geometry and mathematical physics through its role in K-theory. By providing a means to translate topological questions into geometric language, this character helps uncover deep relationships between topological invariants and physical theories. Its applications in index theory, as well as its influence on the development of new mathematical techniques, highlight its importance as a unifying concept in modern mathematics.

Related terms

Chern Character: A fundamental invariant in topology that associates a characteristic class to vector bundles, providing a way to study their properties through cohomology.

K-theory: A branch of algebraic topology that studies vector bundles and their classes using algebraic techniques, specifically focusing on K-groups and their applications.

Cohomology: A mathematical tool used to study topological spaces through algebraic structures, allowing for the analysis of shapes and their properties.