Characteristic classes of foliations

Definition

Characteristic classes of foliations are topological invariants that help describe the geometric properties of foliated manifolds. They are essential for understanding how a foliation interacts with the underlying manifold, providing insights into the global structure and classification of the foliation. These classes relate to the curvature and other geometric features of the leaves of the foliation, enabling mathematicians to analyze and compare different foliations.

5 Must Know Facts For Your Next Test

1. Characteristic classes of foliations can be computed using Chern-Weil theory, which connects differential geometry with algebraic topology.
2. These classes help distinguish different foliated structures on a manifold, as they capture essential geometric information.
3. They are particularly useful in studying transversality properties and singularities within foliated manifolds.
4. The behavior of characteristic classes under deformation provides insights into how foliations can change while preserving certain properties.
5. Characteristic classes can also be related to other invariants, such as the signature of the manifold, offering a richer understanding of its topology.

Review Questions

• How do characteristic classes of foliations contribute to our understanding of foliated manifolds?
  • Characteristic classes of foliations serve as important tools for capturing and describing the geometric properties of foliated manifolds. By providing topological invariants, they help classify different types of foliations and reveal how these structures interact with the underlying manifold. This understanding is crucial when analyzing properties such as curvature, transversality, and singularities within the leaves.
• Discuss the relationship between characteristic classes and Chern classes in the context of foliations.
  • Characteristic classes of foliations are closely related to Chern classes, as both are topological invariants that provide information about vector bundles over manifolds. Chern classes can be used to compute characteristic classes for foliations through Chern-Weil theory. This connection allows for a deeper exploration into the geometric structure of foliations by leveraging the rich properties associated with Chern classes.
• Evaluate the implications of characteristic classes on the study of singularities in foliated manifolds.
  • Characteristic classes play a critical role in understanding singularities in foliated manifolds by offering insights into their topological and geometric nature. The ability to compute these classes allows researchers to identify when singularities arise and how they affect the overall structure of the foliation. By evaluating these implications, mathematicians can determine stability conditions and study how perturbations might change the foliation while maintaining certain characteristics, thus enhancing our comprehension of complex geometrical phenomena.

Related terms

Foliation: A partition of a manifold into disjoint subsets called leaves, which are typically smooth submanifolds.

Chern Classes: Topological invariants associated with vector bundles that provide important information about the geometry of the underlying manifold.

Euler Class: A characteristic class that measures the obstruction to finding non-vanishing sections of a vector bundle, related to the topology of the manifold.