Cohomology
Introduction
In the field of mathematics, particularly in algebraic topology, cohomology is a general term for a sequence of abelian groups associated with a topological space. It is an essential tool in distinguishing topological spaces up to homeomorphism, and it also plays a vital role in formulating the intersection theory of manifolds.
Definition
Cohomology can be defined using the language of category theory. Given a category of topological spaces, one can define a cohomology theory as a functor from this category to the category of graded abelian groups, satisfying certain axioms. These axioms, known as the Eilenberg-Steenrod axioms, ensure that the cohomology theory behaves in a way that is useful for studying topological spaces.
Types of Cohomology
There are several types of cohomology theories, each with its own set of properties and applications. Some of the most important types include:
Singular Cohomology
Singular cohomology is the most basic and fundamental type of cohomology. It is defined using singular chains, which are formal sums of continuous maps from the standard n-simplex to the topological space.
Cech Cohomology
Cech cohomology is a type of cohomology theory that is particularly useful in the study of sheaves and bundles. It is defined using the concept of a Cech cover, which is a collection of open sets that cover the topological space.
De Rham Cohomology
De Rham cohomology is a type of cohomology theory that is particularly useful in the study of differential forms and manifolds. It is defined using the exterior derivative, which is a differential operator that acts on differential forms.
Sheaf Cohomology
Sheaf cohomology is a type of cohomology theory that is particularly useful in the study of sheaves and algebraic geometry. It is defined using the concept of a sheaf, which is a tool for systematically tracking locally defined data attached to the open sets of a topological space.
Applications
Cohomology has a wide range of applications in various areas of mathematics. Some of the most notable applications include:
Topological Invariants
Cohomology groups serve as topological invariants, which are tools used in the classification of topological spaces. Two topological spaces are said to be homotopy equivalent if their cohomology groups are isomorphic.
Intersection Theory
Cohomology plays a crucial role in the formulation of the intersection theory of manifolds. This theory, which studies the intersection of submanifolds of a given manifold, is a fundamental tool in algebraic geometry and topology.
Gauge Theory
In gauge theory, a branch of theoretical physics, cohomology appears in the study of gauge transformations and the classification of gauge fields.