Cobordism theory

Definition

Cobordism theory is a branch of topology that studies the relationships between manifolds through the notion of cobordism, where two manifolds are considered equivalent if there exists a manifold whose boundary is formed by those two manifolds. This theory links various topological invariants and plays a crucial role in classifying manifolds, particularly in terms of their dimensions and structures. It connects with important concepts such as characteristic classes and helps in understanding how different mathematical objects can be related or transformed into one another.

5 Must Know Facts For Your Next Test

1. Cobordism is an equivalence relation among manifolds, meaning that two manifolds are cobordant if they can be the boundary of a third manifold.
2. The study of cobordism led to the development of Wu classes, which are used to derive invariants associated with the cobordism groups.
3. In terms of Stiefel-Whitney classes, cobordism theory provides tools to understand how these classes can change under cobordant transformations of manifolds.
4. Adem relations arise in the context of cobordism as they provide relations among characteristic classes and help simplify calculations involving these classes.
5. Cobordism groups can be classified into different types, such as oriented cobordism and spin cobordism, each with its own set of invariants.

Review Questions

• How does cobordism theory relate to the classification of manifolds and their topological properties?
  • Cobordism theory is essential for classifying manifolds by examining the relationships between them through equivalence under cobordism. Two manifolds are considered equivalent if there exists a third manifold whose boundary comprises both. This relationship allows for an organized way to group manifolds based on their structures and dimensions, helping mathematicians understand the overall landscape of topological spaces.
• Discuss the importance of Wu classes in relation to cobordism theory and their application in determining manifold characteristics.
  • Wu classes are vital in cobordism theory as they provide cohomological invariants that help distinguish between different manifolds. These classes arise naturally when studying how certain properties of manifolds can change under cobordant transformations. By leveraging Wu classes, mathematicians can gain deeper insights into the invariants associated with cobordism groups, facilitating a better understanding of manifold classification.
• Evaluate the role of Adem relations in simplifying calculations within cobordism theory and its implications for characteristic classes.
  • Adem relations play a significant role in simplifying calculations related to characteristic classes within the framework of cobordism theory. They offer essential algebraic identities that connect various characteristic classes and facilitate computations involving them. By utilizing these relations, mathematicians can more effectively analyze the properties of manifolds and their relationships, leading to broader insights into the connections between different topological concepts and paving the way for advances in topology.

Related terms

Manifold: A manifold is a topological space that locally resembles Euclidean space and can be described by charts or coordinate systems.

Characteristic Classes: Characteristic classes are cohomology classes associated with vector bundles that provide invariants to classify bundles over a manifold.

Homotopy: Homotopy is a concept in topology that refers to a continuous deformation of one function into another, providing insights into the shape and structure of spaces.