Cobordism

Definition

Cobordism is a relation between manifolds that captures the idea of one manifold being the boundary of another. More formally, two manifolds are said to be cobordant if there exists a manifold whose boundary consists of the two given manifolds. This concept is crucial in understanding various topological invariants and has significant implications in the study of dualities and classifications within topology.

5 Must Know Facts For Your Next Test

1. Cobordism forms an equivalence relation on the set of manifolds, grouping them into cobordism classes based on whether they can be connected through a common boundary.
2. The cobordism relation is a central theme in the Thom isomorphism theorem, which connects cobordism theory to cohomology theories, revealing deep connections between topology and algebra.
3. In Morse theory, cobordism helps understand how critical points of a function define the topology of level sets and provides insights into how manifolds change as parameters vary.
4. Cobordism also plays a role in string theory and theoretical physics, where it can represent dualities between different physical theories and provide insights into their geometrical structures.
5. The study of cobordism has led to the development of various invariants that classify manifolds, such as stable homotopy groups, which are essential in modern algebraic topology.

Review Questions

• How does cobordism serve as an equivalence relation among manifolds, and why is this property important?
  • Cobordism serves as an equivalence relation by grouping manifolds that can be connected through a single higher-dimensional manifold whose boundary consists of the two original manifolds. This property is important because it allows mathematicians to classify and compare different types of manifolds based on their boundaries, leading to a deeper understanding of their topological properties. It also helps reveal connections between seemingly unrelated manifolds through their shared boundaries.
• Discuss the relationship between cobordism and the Thom isomorphism theorem, highlighting its significance in algebraic topology.
  • The relationship between cobordism and the Thom isomorphism theorem lies in how both concepts connect the geometry of manifolds to algebraic structures. The Thom isomorphism theorem states that cobordism classes correspond to cohomology classes in a way that allows for powerful computations. This connection not only provides tools for classifying manifolds but also shows how topological features manifest in algebraic terms, ultimately bridging the gap between topology and algebra.
• Evaluate the implications of cobordism in Morse theory, particularly regarding critical points and level sets.
  • In Morse theory, cobordism has significant implications as it relates critical points of smooth functions to the topology of their level sets. Each critical point corresponds to changes in the topology as one moves through levels defined by the function's values. The concept of cobordism allows for understanding how these changes can be captured in a broader context, showing how manifolds can transition from one shape to another through smooth deformations. This insight enhances our understanding of manifold behavior under perturbations and reveals intricate relationships between topology and analysis.

Related terms

manifold: A manifold is a topological space that locally resembles Euclidean space and can be described by coordinates, making it possible to perform calculus and other mathematical operations.

bordism: Bordism is a specific type of cobordism that deals with smooth manifolds and emphasizes differentiable structures, allowing for the classification of manifolds based on their smoothness properties.

homology: Homology is a mathematical concept that associates sequences of abelian groups or modules with a topological space, capturing information about its shape and structure through algebraic means.