Bordism

Definition

Bordism is a concept in algebraic topology that studies the relationship between manifolds through the idea of cobordism. It involves classifying manifolds based on whether they can be connected by a 'cobordism,' which is essentially a higher-dimensional manifold that serves as a boundary between two lower-dimensional manifolds. This idea links closely to other topological concepts, such as homotopy and the study of smooth structures on manifolds.

5 Must Know Facts For Your Next Test

1. Bordism provides a framework for understanding how different manifolds can be considered equivalent under cobordism relations.
2. In bordism theory, two manifolds are considered equivalent if there exists a cobordism connecting them, highlighting their topological similarities.
3. Bordism classes can be studied using various invariants, which help classify manifolds in a more manageable way.
4. The concept of bordism extends to both oriented and unoriented manifolds, leading to different types of bordism theories.
5. Bordism has applications in various fields, including algebraic topology, differential geometry, and even mathematical physics.

Review Questions

• How does the concept of bordism connect manifolds and what role does cobordism play in this relationship?
  • Bordism connects manifolds by defining an equivalence relation based on whether they can be linked by a cobordism, which is a higher-dimensional manifold that serves as the boundary between the two. This means that if two manifolds have a cobordism, they are considered equivalent in the context of bordism theory. Cobordisms essentially allow us to visualize and understand the relationships between different manifolds by providing a geometric structure that connects them.
• Discuss the significance of bordism classes and how they can be used to study topological invariants.
  • Bordism classes are significant because they group together manifolds that are equivalent under cobordism, allowing mathematicians to focus on these classes rather than individual manifolds. This classification leads to the identification of topological invariants that characterize these classes, making it easier to study their properties. For example, by analyzing these invariants, one can determine whether two seemingly different manifolds belong to the same bordism class and thus share fundamental topological characteristics.
• Evaluate the broader implications of bordism in algebraic topology and its influence on modern mathematical theories.
  • Bordism has broader implications in algebraic topology as it provides a unifying framework for understanding various topological phenomena. Its influence extends to modern mathematical theories, such as string theory and quantum field theory, where concepts of manifold boundaries are crucial. By exploring how manifolds relate through bordism, researchers gain insights into the structure of spaces in higher dimensions, influencing both theoretical developments and practical applications in physics and geometry.

Related terms

Cobordism: A relation between two manifolds where a manifold serves as a boundary, effectively linking them within a higher-dimensional space.

Homotopy: A concept in topology that studies when two continuous functions can be transformed into each other through continuous deformations.

Manifold: A topological space that locally resembles Euclidean space and can be studied using calculus and algebraic methods.