Pontryagin numbers

Definition

Pontryagin numbers are topological invariants associated with smooth manifolds, particularly in the study of cobordism theory. They arise from the Pontryagin classes, which are characteristic classes that help in understanding the topology of vector bundles over manifolds. These numbers provide crucial information about the manifold's differential structure and are used to classify manifolds up to cobordism.

5 Must Know Facts For Your Next Test

1. Pontryagin numbers are calculated from the Pontryagin classes of a manifold and can be expressed as integrals over certain combinations of these classes.
2. The first Pontryagin number is closely related to the signature of the manifold, linking topology to algebraic invariants.
3. These numbers are used to define the Pontryagin ring, which encodes information about the manifold's topology and interacts with other characteristic classes.
4. Pontryagin numbers play a vital role in distinguishing manifolds that cannot be smoothly deformed into each other, showcasing their significance in cobordism theory.
5. In dimensions where they are defined, Pontryagin numbers can provide nontrivial obstructions to certain types of smooth structures on manifolds.

Review Questions

• How do Pontryagin numbers relate to the classification of manifolds in cobordism theory?
  • Pontryagin numbers serve as topological invariants that help distinguish between different smooth structures on manifolds. In cobordism theory, two manifolds are considered equivalent if they can be transformed into each other through continuous deformations. The values of Pontryagin numbers can differ between these structures, indicating that they cannot be smoothly deformed into one another, thus providing critical insights into their classification.
• Discuss how Pontryagin classes contribute to the computation of Pontryagin numbers and their implications in topology.
  • Pontryagin classes are characteristic classes associated with real vector bundles that generate Pontryagin numbers when integrated over the manifold. These classes provide a way to encode information about the manifold's curvature and topology. The computation of Pontryagin numbers from these classes allows mathematicians to derive important topological properties of manifolds, influencing their classification within cobordism theory and aiding in understanding their geometric structures.
• Evaluate the importance of Pontryagin numbers in establishing obstructions to smooth structures on manifolds and how this impacts cobordism theory.
  • Pontryagin numbers are crucial for identifying obstructions to certain smooth structures on manifolds, meaning they can indicate whether a manifold can support a particular type of smooth structure. This has significant implications in cobordism theory, where understanding which manifolds can or cannot be smoothly connected informs broader classification efforts. By evaluating these obstructions, mathematicians can gain insights into the complexities of manifold topology and develop better frameworks for understanding high-dimensional shapes.

Related terms

Characteristic classes: Characteristic classes are a way to associate a cohomology class to a vector bundle, which helps in distinguishing different bundles and studying their properties.

Cohomology:

Cohomology is a mathematical tool used to study topological spaces through algebraic invariants, allowing the classification and analysis of their structure.

Cobordism: Cobordism is a relation between manifolds where two manifolds are considered equivalent if one can be transformed into the other through a continuous deformation, providing a framework for classifying manifolds.