Stiefel-Whitney classes

Definition

Stiefel-Whitney classes are a set of characteristic classes associated with real vector bundles, providing important invariants that capture the topology of the underlying manifold. These classes help to classify vector bundles over a space and can reveal properties such as orientability and the existence of certain types of sections. They play a crucial role in connecting various concepts like fibrations, vector bundles, and their applications in manifold theory.

5 Must Know Facts For Your Next Test

1. Stiefel-Whitney classes are denoted by \( w_i(E) \), where \( E \) is a vector bundle and \( i \) indicates the class's degree.
2. The first Stiefel-Whitney class \( w_1(E) \) determines whether the bundle is orientable, being zero if the bundle is orientable.
3. Stiefel-Whitney classes can be computed using the total Stiefel-Whitney class, which is a generating function for all Stiefel-Whitney classes.
4. These classes satisfy several important properties such as additivity for direct sums of vector bundles and multiplicative relations when considering products.
5. They also have significant applications in cobordism theory, helping to determine the relationships between different manifolds based on their characteristic classes.

Review Questions

• How do Stiefel-Whitney classes relate to the concept of vector bundles and what significance do they have?
  • Stiefel-Whitney classes are vital for understanding vector bundles as they provide topological invariants that classify these bundles. Specifically, they help determine properties such as orientability and sections of the bundle. For instance, if the first Stiefel-Whitney class is zero, it indicates that the vector bundle is orientable. This classification aids in examining how vector bundles behave over different manifolds.
• Discuss how Stiefel-Whitney classes interact with fibrations and why this relationship is important in topology.
  • Stiefel-Whitney classes play an essential role in the study of fibrations since they can be used to understand how fiber bundles behave under various topological conditions. The lifting properties inherent to fibrations often allow for a clearer analysis of sections and homotopy types. By examining the Stiefel-Whitney classes of the fibers in a fibration, one can derive insights about the total space and base space topology.
• Evaluate the implications of Stiefel-Whitney classes in relation to manifold classification and cobordism theory.
  • Stiefel-Whitney classes significantly impact manifold classification by providing necessary invariants for distinguishing between different manifolds. In cobordism theory, these classes help identify relationships between manifolds by comparing their characteristic classes. Two manifolds with identical Stiefel-Whitney classes may have similar topological structures, indicating potential cobordism. This relationship showcases how characteristic classes can be powerful tools in understanding the deeper connections within manifold theory.

Related terms

Characteristic classes: Invariants that classify fiber bundles, revealing topological properties of the underlying space.

Vector bundles: A collection of vector spaces parametrized by a topological space, where each point in the base space has an associated vector space.

Fibration:

A special type of fiber bundle that satisfies certain lifting properties, important in studying homotopy and homology.