Godbillon-vey class

Definition

The godbillon-vey class is an important invariant associated with foliated manifolds, which measures the twisting of the leaves of a foliation. This class arises from the study of connections on the tangent bundles of manifolds and helps to classify foliations in terms of their geometric properties. It provides insights into the topology of the manifold by linking the structure of foliations with characteristic classes, such as the Chern classes.

5 Must Know Facts For Your Next Test

1. The godbillon-vey class can be defined in terms of the holonomy of a foliation, reflecting how the leaves twist around each other.
2. It is a specific type of secondary characteristic class that exists in dimensions three or higher.
3. The godbillon-vey class vanishes if the foliation is simply reducible or if it admits a transverse measure.
4. This invariant plays a critical role in understanding the interplay between foliations and dynamical systems.
5. In the context of foliated manifolds, the godbillon-vey class provides essential information about the topology and geometry of the manifold as a whole.

Review Questions

• How does the godbillon-vey class relate to the concept of foliation and what does it indicate about the leaves?
  • The godbillon-vey class is directly tied to foliations as it measures how the leaves twist and interact within the manifold. It indicates whether a foliation has complex structure or if it is simpler, such as being reducible. By studying this invariant, one can gain insights into how the geometric arrangement of leaves affects the overall topology of the manifold.
• Discuss the significance of characteristic classes in relation to the godbillon-vey class and foliated manifolds.
  • Characteristic classes provide essential algebraic tools for classifying vector bundles and understanding their topological properties. The godbillon-vey class is considered a secondary characteristic class that specifically applies to foliated manifolds, connecting it to broader topological concepts. This relationship highlights how various geometric structures can influence each other, providing deeper insights into both foliation theory and characteristic classes.
• Evaluate how the vanishing of the godbillon-vey class can impact our understanding of foliations and their properties.
  • When the godbillon-vey class vanishes, it indicates that a foliation is simpler than initially perceived, such as being reducible or admitting transverse measures. This vanishing has significant implications for studying dynamical systems and foliated structures, suggesting a more straightforward behavior among the leaves. Analyzing cases where this class vanishes allows researchers to identify special characteristics within foliated manifolds and explore their implications for topology and geometry.

Related terms

Foliation: A partition of a manifold into a collection of disjoint submanifolds, called leaves, which resemble lower-dimensional manifolds.

Characteristic Class: An algebraic invariant that classifies vector bundles over a manifold, providing crucial information about the topology of the underlying space.

Tangential Structure: The geometric framework associated with the tangent spaces of a manifold, which is essential for understanding differentiable structures and their properties.