Cobordism
Introduction
A cobordism is a fundamental concept in the field of topology, specifically in the study of manifolds. It is a mathematical tool used to compare different manifolds and serves as a bridge between them. The concept of cobordism was first introduced by Lev Pontryagin in the 1940s and has since been a vital part of manifold theory and its applications.
Definition
Formally, a cobordism between two manifolds M and N is a manifold W, along with embeddings of M and N into the boundary of W, such that the boundary of W is the disjoint union of the images of M and N. This is often visualized as M and N being "ends" of W, with W serving as a sort of "bridge" or "corridor" between M and N.
Cobordism Classes
Cobordism forms an equivalence relation on the set of all manifolds, and the equivalence classes under this relation are called cobordism classes. Two manifolds are said to be cobordant if they belong to the same cobordism class. The set of all cobordism classes forms a group under the operation of disjoint union, known as the cobordism group.
Cobordism Group
The cobordism group is an important object of study in manifold theory. It is a group whose elements are cobordism classes, and whose group operation is given by disjoint union. The identity element is the class of the empty set, and the inverse of a class is the class itself. The cobordism group is often denoted by Ω.
Oriented and Unoriented Cobordism
Cobordism can be either oriented or unoriented. In the case of oriented cobordism, the manifolds involved are assumed to be oriented, and the cobordism is required to preserve this orientation. In unoriented cobordism, no such assumption is made. The study of oriented cobordism leads to the concept of the oriented cobordism group, denoted by ΩO, while unoriented cobordism leads to the unoriented cobordism group, denoted by ΩU.
Cobordism in Algebraic Topology
Cobordism plays a significant role in algebraic topology, particularly in the study of characteristic classes. It provides a way to define and study cohomology theories, leading to the concept of cobordism theories. These theories have been instrumental in solving many problems in algebraic topology and have found applications in other areas of mathematics as well.
Cobordism in Physics
In physics, cobordism finds application in the field of quantum field theory. In particular, the concept of cobordism is used in the formulation of topological quantum field theories, where it provides a way to define and study the path integral. This has led to deep connections between physics and topology, and has been a fruitful area of research.