Cobordant manifolds

Definition

Cobordant manifolds are pairs of manifolds that represent the same class in a cobordism relation, which intuitively means that they can be 'glued' together to form a higher-dimensional manifold with boundary. This relationship is essential in topology as it allows the study of different manifolds through their connections and transformations, particularly in understanding how they can be related via homotopy and other topological features.

5 Must Know Facts For Your Next Test

1. Two manifolds are said to be cobordant if their disjoint union is the boundary of some higher-dimensional manifold.
2. Cobordism classes can be thought of as equivalence classes where manifolds are equivalent if they are cobordant to each other.
3. The h-cobordism theorem states that if two smooth manifolds are cobordant and certain conditions are met, they are diffeomorphic, meaning they can be smoothly transformed into one another.
4. In terms of dimensions, if $M$ and $N$ are cobordant manifolds, then their dimensions must match except for the addition of one dimension in the higher-dimensional manifold formed by their union.
5. The concept of cobordism is a critical tool in algebraic topology, as it connects various invariants and helps classify manifolds up to certain topological properties.

Review Questions

• How do cobordant manifolds relate to the concept of boundaries in topology?
  • Cobordant manifolds are intimately related to boundaries because they can be combined to create a new manifold whose boundary is formed from the original pair. Specifically, if two manifolds are cobordant, their disjoint union serves as the boundary of a higher-dimensional manifold. This relationship illustrates how boundaries help define the structure and connections between different manifolds in topology.
• Discuss the implications of the h-cobordism theorem on the classification of smooth manifolds.
  • The h-cobordism theorem has significant implications for classifying smooth manifolds by establishing that if two cobordant smooth manifolds meet specific criteria, they are diffeomorphic. This means that under certain conditions, cobordism not only indicates a topological relationship but also implies that these manifolds share very similar geometric properties. As a result, this theorem allows mathematicians to focus on cobordism classes when studying the topology of smooth manifolds.
• Evaluate the importance of cobordism in understanding the relationships between different topological spaces and their invariants.
  • Cobordism plays a crucial role in understanding the relationships between different topological spaces by providing a framework to relate them through equivalence classes. By studying cobordant manifolds, mathematicians can uncover deeper insights into their topological invariants and how these invariants behave under transformations. This evaluation highlights how cobordism not only aids in classification but also enhances our comprehension of the rich interplay between geometry and topology in mathematical analysis.

Related terms

Manifold:

A manifold is a topological space that locally resembles Euclidean space, allowing for calculus and other analytical techniques to be applied.

Boundary: The boundary of a manifold is the set of points that can be approached from within the manifold but do not belong to it, defining the limits of the manifold.

Homotopy:

Homotopy is a concept in topology that deals with the continuous transformation of one function into another, providing a way to classify spaces based on their shapes and structures.