Cobordism invariant

Definition

A cobordism invariant is a property of manifolds that remains unchanged under cobordism, meaning that two manifolds are considered equivalent if there exists a higher-dimensional manifold whose boundary consists of those two manifolds. This concept helps to classify and relate different manifolds by determining which properties can be preserved in the cobordism relation.

5 Must Know Facts For Your Next Test

1. Cobordism invariants can be used to distinguish between different types of manifolds, such as orientable and non-orientable manifolds.
2. The study of cobordism invariants is closely related to the concept of characteristic classes, which provide additional algebraic structures to help understand manifold properties.
3. Certain invariants, like the Euler characteristic or signature, can serve as cobordism invariants, aiding in classification tasks in topology.
4. Cobordism is an equivalence relation, meaning it partitions manifolds into equivalence classes based on the existence of cobordisms between them.
5. Cobordism theory has significant implications in various fields such as algebraic topology, differential topology, and even theoretical physics.

Review Questions

• How do cobordism invariants assist in classifying different types of manifolds?
  • Cobordism invariants help classify manifolds by providing properties that remain unchanged under cobordism relations. For instance, two manifolds may be considered equivalent if they can be connected through a higher-dimensional manifold. This allows mathematicians to group manifolds into equivalence classes based on shared properties, making it easier to study their relationships and characteristics.
• Discuss the relationship between cobordism invariants and characteristic classes.
  • Cobordism invariants are deeply connected to characteristic classes, which serve as additional algebraic structures that describe the properties of vector bundles over manifolds. While cobordism invariants focus on the topological aspects of manifolds and their boundaries, characteristic classes provide a way to understand more intricate features like curvature and torsion. Together, they enrich the study of manifold properties and enhance our understanding of their classification.
• Evaluate the impact of cobordism theory on modern mathematical research and its applications beyond topology.
  • Cobordism theory has significantly influenced modern mathematical research by introducing new frameworks for understanding manifold relationships. Its applications extend beyond pure mathematics into fields like theoretical physics, where it informs concepts in quantum field theory and string theory. The tools and ideas from cobordism theory continue to inspire advancements in algebraic topology, geometry, and even mathematical physics, showcasing its importance in shaping contemporary mathematical thought.

Related terms

Manifold: A topological space that locally resembles Euclidean space and is defined by its dimension, allowing for calculus and geometric concepts to be applied.

Cobordism: A relation between two manifolds where they can be connected through a third manifold, serving as their boundary, essentially forming a higher-dimensional 'bridge' between them.

Homotopy: A concept in algebraic topology that deals with the deformation of one continuous function into another, allowing for the classification of topological spaces based on their fundamental shapes.