Cobordism

Definition

Cobordism is a concept in topology that relates two manifolds through a higher-dimensional manifold, called a cobordism, that connects them. This idea is fundamental in understanding how manifolds can be transformed into one another and provides a powerful tool for classifying manifolds based on their boundaries and the relationships between them.

5 Must Know Facts For Your Next Test

1. Cobordism provides a way to classify manifolds based on their boundaries; two manifolds are cobordant if there exists a third manifold whose boundary consists of the two original manifolds.
2. The study of cobordism led to the development of important invariants in topology, such as cobordism groups, which can reveal deep relationships between different topological spaces.
3. In Morse theory, cobordism helps understand how critical points of Morse functions correspond to changes in the topology of the manifold as it evolves.
4. Cobordism is closely tied to homotopy theory, as it allows the comparison of manifold structures under continuous transformations.
5. The h-cobordism theorem states that if two simply connected smooth manifolds are cobordant, then they are homeomorphic, showing a strong link between cobordism and the structure of manifolds.

Review Questions

• How does cobordism relate to the classification of manifolds and their boundaries?
  • Cobordism plays a crucial role in classifying manifolds by establishing a relationship between their boundaries. Two manifolds are said to be cobordant if there exists another manifold whose boundary consists of the two given manifolds. This connection allows mathematicians to group manifolds based on shared properties and understand their topological features more deeply.
• Discuss the implications of Morse functions on cobordisms and how they can affect critical points.
  • Morse functions provide insight into the topology of a manifold through its critical points, and this understanding extends to cobordisms. When analyzing a cobordism with respect to Morse functions, the critical points represent changes in the topology as one transitions from one manifold to another. The way these critical points interact with each other can illustrate how the topology alters during this process, highlighting important features of both manifolds involved.
• Evaluate the significance of the h-cobordism theorem in the study of smooth manifolds and its implications for topological equivalence.
  • The h-cobordism theorem is significant because it establishes that if two simply connected smooth manifolds are cobordant, they must also be homeomorphic. This result connects cobordism directly with topological equivalence, demonstrating that such structures are not only related but fundamentally identical in terms of their topological properties. This finding has far-reaching implications in topology and geometry, affecting how mathematicians approach the study of manifold structures and their transformations.

Related terms

Manifold: A manifold is a topological space that locally resembles Euclidean space and can be described by coordinates. They serve as the primary objects of study in topology.

Homotopy: Homotopy is a concept that describes when two continuous functions can be continuously transformed into each other. It's essential for understanding the relationships between different topological spaces.

Surgery Theory: Surgery theory is a method in topology used to modify manifolds by removing and replacing parts of them. It provides tools for understanding how manifolds can be manipulated in relation to cobordism.