Non-abelian cohomology/

Introduction

Non-abelian cohomology is a branch of mathematics that extends the concepts of abelian cohomology to non-abelian settings. It is a rich and complex field that finds applications in various areas such as algebraic geometry, topology, and number theory. Unlike abelian cohomology, where the cohomology groups are abelian, non-abelian cohomology deals with more general structures, often leading to non-abelian groups or even more complex algebraic structures.

Historical Background

The development of non-abelian cohomology can be traced back to the mid-20th century, with significant contributions from mathematicians such as Jean-Pierre Serre, Alexander Grothendieck, and Michael Artin. The initial motivation was to generalize the classical cohomology theories to settings where the underlying structures are not necessarily commutative. This led to the development of various tools and techniques that are now fundamental in the study of non-abelian cohomology.

Basic Concepts

Non-abelian Groups

In non-abelian cohomology, the primary objects of study are non-abelian groups. Unlike abelian groups, where the group operation is commutative, non-abelian groups do not satisfy this property. This lack of commutativity introduces additional complexity in the cohomological analysis.

Cohomology Sets

One of the key differences between abelian and non-abelian cohomology is that the latter often deals with cohomology sets rather than cohomology groups. These sets can have more intricate structures and are not necessarily groups themselves. For example, the first non-abelian cohomology set \(H^1(G, A)\) for a group \(G\) acting on a non-abelian group \(A\) is a pointed set rather than a group.

Non-abelian Extensions

Non-abelian cohomology is closely related to the study of non-abelian extensions. Given a group \(G\) and a non-abelian group \(A\), a non-abelian extension of \(G\) by \(A\) is a group \(E\) such that there is an exact sequence: \[ 1 \to A \to E \to G \to 1 \] The classification of such extensions is a central problem in non-abelian cohomology.

Non-abelian Cocycles and Coboundaries

In non-abelian cohomology, cocycles and coboundaries play a crucial role. Given a group \(G\) acting on a non-abelian group \(A\), a 1-cocycle is a function \(f: G \to A\) satisfying a specific condition. Similarly, coboundaries are functions that can be expressed in a particular form. The set of 1-cocycles modulo coboundaries forms the first non-abelian cohomology set \(H^1(G, A)\).

Applications

Non-abelian cohomology has numerous applications in various fields of mathematics. In algebraic geometry, it is used to study the classification of principal bundles and torsors. In topology, it helps in understanding the structure of fiber bundles and covering spaces. In number theory, non-abelian cohomology plays a role in the study of Galois cohomology and the arithmetic of algebraic groups.

Advanced Topics

Higher Non-abelian Cohomology

While the first non-abelian cohomology set \(H^1(G, A)\) is well-studied, higher non-abelian cohomology sets \(H^n(G, A)\) for \(n > 1\) are more complex and less understood. These higher cohomology sets often involve more intricate algebraic structures and require advanced techniques for their study.

Non-abelian Sheaf Cohomology

In the context of sheaf theory, non-abelian cohomology is used to study sheaves of non-abelian groups. This involves generalizing the classical sheaf cohomology to settings where the stalks of the sheaves are non-abelian groups. This area has deep connections with algebraic topology and complex geometry.

References