Stiefel-Whitney classes

Definition

Stiefel-Whitney classes are characteristic classes associated with real vector bundles, providing important topological invariants that help classify these bundles. They play a significant role in the study of manifold properties, particularly in relation to cohomology theories, where they reveal information about the intersection of submanifolds and the topology of vector bundles. These classes are particularly useful for understanding orientability and the existence of certain structures on manifolds.

5 Must Know Facts For Your Next Test

1. Stiefel-Whitney classes are denoted as $w_i(E)$ for a vector bundle $E$, where $i$ represents the index of the class.
2. The first Stiefel-Whitney class $w_1(E)$ indicates whether a vector bundle is orientable; if it is non-zero, the bundle cannot be oriented.
3. These classes are computed using cohomology, specifically in mod 2 cohomology, leading to interesting results in topology and geometry.
4. The total Stiefel-Whitney class of a vector bundle can be expressed as $w(E) = 1 + w_1(E) + w_2(E) + ...$, representing all the characteristic classes associated with the bundle.
5. Stiefel-Whitney classes relate to Wu classes through Wu's formula, which connects them with specific operations in cohomology.

Review Questions

• How do Stiefel-Whitney classes contribute to our understanding of orientability in vector bundles?
  • Stiefel-Whitney classes help determine whether a vector bundle is orientable by examining the first class $w_1(E)$. If $w_1(E)$ is non-zero, it indicates that the vector bundle cannot be oriented, meaning there is no consistent choice of direction across the entire bundle. This connection highlights the importance of these classes in distinguishing between different types of vector bundles based on their geometric properties.
• Describe the significance of the total Stiefel-Whitney class in the context of characteristic classes.
  • The total Stiefel-Whitney class provides a comprehensive representation of all Stiefel-Whitney classes associated with a vector bundle. It is expressed as $w(E) = 1 + w_1(E) + w_2(E) + ...$, where each term contributes information about different aspects of the bundle's topology. This total class encapsulates essential features and invariants that can help classify vector bundles and understand their geometric behavior within a manifold.
• Evaluate the relationship between Stiefel-Whitney classes and Wu classes, focusing on their roles in cohomology theories.
  • Stiefel-Whitney classes and Wu classes are closely related in that they both serve as characteristic classes used to study vector bundles through cohomology theories. Wu's formula establishes a connection between these two types of classes, demonstrating how Wu classes can be derived from Stiefel-Whitney classes using specific operations in cohomology. This relationship deepens our understanding of how various topological invariants interact and reinforces the utility of these classes in studying complex topological structures.

Related terms

Characteristic classes: Topological invariants that classify vector bundles over a space, helping to distinguish between different types of bundles based on their geometric properties.

Cohomology: A mathematical tool for studying topological spaces by associating algebraic structures, such as groups or rings, with these spaces, often used to compute invariants like Stiefel-Whitney classes.

Vector bundles: Mathematical structures that consist of a family of vector spaces parameterized continuously by a topological space, allowing for the study of functions and sections defined on them.