Cobordism

Definition

Cobordism is a concept in topology that studies the relationship between manifolds through the idea of boundaries. Essentially, two manifolds are cobordant if they can be connected by a higher-dimensional manifold whose boundary is the disjoint union of the two manifolds. This notion plays a critical role in understanding how manifolds relate to each other and has important implications in fields such as surgery theory and L-theory.

5 Must Know Facts For Your Next Test

1. In cobordism, the key idea is that if two manifolds are cobordant, they share a common 'cobordism' which acts like a bridge connecting them in a higher dimension.
2. Cobordism is an equivalence relation, meaning it satisfies reflexivity, symmetry, and transitivity, allowing for a structured way to classify manifolds.
3. The study of cobordism can reveal deep insights into the properties of manifolds, such as their dimensions and types of singularities they may have.
4. Cobordism theories often relate to characteristic classes, which are algebraic invariants that provide additional information about the manifold's topology.
5. In surgery theory, cobordism helps identify when two manifolds can be considered equivalent, significantly simplifying the classification of high-dimensional topological spaces.

Review Questions

• How does the concept of cobordism enhance our understanding of the relationships between different manifolds?
  • Cobordism enhances our understanding by providing a framework for relating different manifolds through their boundaries. When two manifolds are found to be cobordant, it indicates that they can be connected through another manifold, which gives insight into their topological similarities and differences. This relationship helps in classifying manifolds and identifying deeper properties that might not be apparent when looking at them individually.
• Discuss how cobordism interacts with surgery theory and its importance in manifold classification.
  • Cobordism interacts closely with surgery theory as it provides criteria for determining when two manifolds can be modified to appear similar through surgery operations. When working within surgery theory, identifying cobordant manifolds allows mathematicians to apply cutting and pasting techniques effectively, aiding in the classification of manifolds. This interplay between the two concepts simplifies complex topological problems by reducing them to more manageable forms.
• Evaluate the implications of cobordism theories on characteristic classes and their role in understanding manifold properties.
  • Cobordism theories have significant implications on characteristic classes as they provide crucial algebraic invariants linked to the topological features of manifolds. Characteristic classes capture essential information about vector bundles over these manifolds, and understanding cobordism helps reveal how changes in the topology of a manifold can affect its characteristic classes. By analyzing these relationships, mathematicians can derive new insights into the structure and classification of manifolds based on their cobordism properties.

Related terms

Manifold: A manifold is a topological space that locally resembles Euclidean space and allows for calculus to be performed on it.

Surgery Theory: Surgery theory is a technique used to modify manifolds by cutting and pasting them in certain ways, often to classify them or to understand their topological properties.

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