Cobordism

Definition

Cobordism is a relation between manifolds where two manifolds are said to be cobordant if there exists a third manifold whose boundary is the disjoint union of the two original manifolds. This concept connects various branches of mathematics, particularly topology and geometry, and plays a vital role in classifying manifolds through their boundaries and their higher-dimensional generalizations.

5 Must Know Facts For Your Next Test

1. Cobordism groups can be classified into different dimensions, with each group representing classes of manifolds that are cobordant to one another.
2. The concept of cobordism extends to higher dimensions, where manifolds can have different topological properties that affect their cobordism classes.
3. There are various types of cobordism, including oriented cobordism and spin cobordism, which impose additional structure on the manifolds involved.
4. Cobordism theory provides tools for calculating invariants in topology, such as the signature and the Euler characteristic, helping to differentiate between non-cobordant manifolds.
5. The study of cobordism has led to significant advancements in understanding the relationships between manifolds and their embeddings in higher-dimensional spaces.

Review Questions

• How does the concept of cobordism relate to the classification of manifolds in topology?
  • Cobordism provides a framework for classifying manifolds based on their boundaries. Two manifolds are considered cobordant if they share a common boundary represented by a third manifold. This relationship helps mathematicians categorize different types of manifolds, revealing deeper connections and structures within topology. The classification simplifies the study of manifold properties by grouping those that can be continuously transformed into one another.
• Discuss the significance of oriented cobordism in the context of cobordism theory and its implications for manifold classification.
  • Oriented cobordism adds an additional layer of structure by considering not just the existence of a cobordant manifold, but also the orientation of the manifolds involved. This means that both original manifolds must have compatible orientations for them to be considered oriented cobordant. This concept is crucial for distinguishing between manifolds that might otherwise appear similar but have fundamentally different properties when orientation is taken into account, thus impacting their classification.
• Evaluate how cobordism theory influences current research in topology and its applications in other areas of mathematics.
  • Cobordism theory has significant implications in modern mathematical research, especially as it relates to algebraic topology and differential geometry. Its ability to classify and connect various types of manifolds aids mathematicians in understanding complex topological spaces and their interactions. Additionally, cobordism concepts find applications in fields such as mathematical physics and string theory, where understanding manifold properties is crucial for formulating theories about the fabric of space-time.

Related terms

manifold:

A manifold is a topological space that locally resembles Euclidean space and can be described by coordinates, allowing for calculus to be performed on it.

homotopy: Homotopy is a concept in topology that describes when two continuous functions can be transformed into each other through continuous deformation.

bordism group: A bordism group is an algebraic structure that classifies manifolds based on the cobordism relation, allowing mathematicians to study the properties and relationships between different manifolds.