Cobordism group

Definition

A cobordism group is an algebraic structure that classifies manifolds based on their boundaries and the existence of cobordisms between them. It reflects the relationship between different manifolds, where two manifolds are considered equivalent if they can be connected by a cobordism, which is essentially a higher-dimensional manifold whose boundary consists of the two manifolds in question. Cobordism groups provide insight into the topology of manifolds and have implications in various areas of mathematics, including algebraic topology and differential topology.

5 Must Know Facts For Your Next Test

1. Cobordism groups are denoted as \(M_n\), where \(n\) represents the dimension of the manifolds being studied.
2. The most common types of cobordism groups are oriented cobordism groups and unoriented cobordism groups, depending on whether orientation is preserved.
3. These groups can be computed using various techniques, including surgery theory and characteristic classes.
4. The rank of a cobordism group can provide information about the number of distinct classes of manifolds within that dimension.
5. Cobordism groups play a crucial role in classifying manifolds and can also connect to other algebraic structures, such as cohomology theories.

Review Questions

• How do cobordism groups facilitate the classification of manifolds?
  • Cobordism groups classify manifolds by establishing equivalence relations based on the existence of cobordisms. Two manifolds are considered equivalent if there is a higher-dimensional manifold connecting them, which allows for a structured way to study their properties. This classification helps mathematicians understand relationships between different types of manifolds and their boundaries.
• What is the difference between oriented and unoriented cobordism groups, and why is this distinction important?
  • Oriented cobordism groups consider manifolds that preserve orientation during the transition through a cobordism, while unoriented cobordism groups do not require this preservation. This distinction is important because it affects the classification and algebraic structure of the groups, leading to different insights into the topological properties of the manifolds involved. The choice between these two types impacts how one can study properties like boundary behavior and manifold intersections.
• Evaluate the role of cobordism groups in connecting to other areas of mathematics, such as cohomology theories.
  • Cobordism groups play a significant role in bridging various mathematical domains by providing tools for studying topological properties that relate to cohomology theories. These connections allow mathematicians to utilize results from algebraic topology to gain insights into manifold classifications. Furthermore, through these relationships, researchers can apply techniques from one area to solve problems in another, highlighting the interconnected nature of modern mathematical theories.

Related terms

manifold: A manifold is a topological space that locally resembles Euclidean space and allows for the extension of concepts such as curvature and dimensions.

cobordism:

Cobordism is a relation between two manifolds, where they are considered cobordant if there exists a manifold whose boundary consists of those two manifolds.

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