Group cohomology

In mathematics, particularly in homological algebra, group cohomology is a powerful set of tools used to study groups by applying techniques from cohomology theory. This theory comes from algebraic topology and helps us understand the structure of groups and how they act on other mathematical objects. Just like group representations, group cohomology examines how a group interacts with something called a G-module, giving us valuable information about both the group and the module.

Group cohomology is closely tied to topology. For a discrete group, its cohomology can be thought of as the singular cohomology of a special space that has the group as its fundamental group. For example, the group cohomology of the integers Z can be viewed as the cohomology of a circle S¹. This connection allows mathematicians to use topological methods to solve problems in group theory.

The study of group cohomology began in the 1920s and grew significantly in the late 1940s. It remains an active area of research today, with many known results about the cohomology of various groups, interpretations of low-dimensional cohomology, and ways to relate the cohomology of different groups. It has applications in abstract algebra, algebraic number theory, and many other fields.

Motivation

In group theory, we often study a group G by looking at its group representations. These representations can be generalized to something called G-modules. A G-module is a special kind of group M where every element of G acts in a certain way on M.

When we look at G-modules, we can find special elements in M that don’t change no matter how G acts on them. These are called invariant elements. Group cohomology helps us understand how these invariant elements behave when we compare different G-modules. It measures how the invariants in one module relate to invariants in another, especially when we look at parts of the module. This gives us deeper insight into the structure of the group G.

Definitions

Group cohomology is a part of mathematics that helps us study groups using ideas from topology. Think of it as a way to understand groups by looking at how they act on other mathematical objects.

Imagine you have a group, and you want to see how it interacts with a special kind of structure called a G-module. Group cohomology gives us tools to explore these interactions. It connects group theory with algebraic topology, a branch of math that studies shapes and spaces.

One way to understand group cohomology is by using something called cochain complexes. These are like step-by-step lists of rules that help us calculate important properties of the group. By studying these rules, we can learn more about the group's structure and behavior.

Group homology

Group homology is a concept in mathematics that studies groups using methods from algebra. It is closely related to group cohomology but focuses on different properties. For a group G and a G-module M, group homology looks at special constructions involving M.

This idea helps mathematicians understand the structure of groups by examining how they interact with other mathematical objects. It uses tools from algebraic topology to explore these relationships in a systematic way.

Low-dimensional cohomology groups

The zero-degree cohomology group shows the module of invariants. The first and second degree cohomologies have simple meanings. The first cohomology group looks at special maps between groups, called crossed homomorphisms, and simplifies when the group action is trivial.

The second cohomology group, when the group action is trivial, corresponds to certain ways of extending a group. For example, it can classify special algebraic structures called division algebras.

Basic examples

Group cohomology is a way to study groups using ideas from geometry and topology. It looks at how a group acts on certain mathematical objects to understand the group's properties better.

One simple example is studying a finite cyclic group, which is a group that repeats itself after a certain number of steps. By examining how this group interacts with numbers, we can compute its cohomology groups — collections of numbers that tell us about the group's structure. Another example involves free groups, which are groups built from combining individual elements without any extra rules. Their cohomology can also be calculated using topological methods, by comparing the group to a space built from circles.

Properties

Group cohomology is a way to study groups using ideas from topology. It looks at how a group acts on certain objects to understand the group's structure better. This helps mathematicians explore the properties of groups by connecting them to shapes and spaces.

One important idea in group cohomology is the "long exact sequence." This sequence shows how the properties of a group and its subgroups are linked. It helps mathematicians calculate important values called cohomology groups, which give information about the group. This sequence is very useful for solving problems in algebra and topology.

Further examples

Group cohomology helps us understand how groups relate to each other, especially in cases where one group controls another in a specific way, called a semi-direct product. This idea uses tools from topology, which is the study of shapes and spaces, to look at groups in a new way.

By thinking of groups as spaces, we can use special sequences and patterns to learn more about their structure. These patterns connect the properties of the larger group to the smaller groups it contains, giving insights into how the group behaves overall.

Cohomology of finite groups

The cohomology groups of finite groups have a special property: they are all torsion for levels one and above. This means that, under certain conditions, these groups become zero. For example, when the order of the group does not divide the characteristics of the field, the cohomology groups vanish.

Tate cohomology combines both homology and cohomology of a finite group. It has useful properties like long exact sequences and product structures. Tate cohomology is especially important in class field theory. For cyclic groups, Tate cohomology shows a repeating pattern every two steps.

Applications

Group cohomology has many useful applications in mathematics. One important area is algebraic K-theory, where it helps us understand the structure of rings and their associated groups. In this context, group cohomology examines how certain sequences of groups stabilize, meaning their properties become predictable after a certain point.

Another key application is in the study of projective representations in quantum mechanics. These are special ways groups can act on spaces, often involving phases or "twists" in their behavior. Group cohomology helps classify these representations by examining the underlying mathematical structures that govern these phases.

Extensions

Group cohomology is a way to study groups using ideas from geometry and algebra. It helps us understand the properties of a group by looking at how it acts on other mathematical objects.

When we deal with groups that have a special kind of structure called a topology, we can use continuous actions to explore their properties further. This leads to areas like Galois cohomology, which is important in number theory. In some cases, we can also study groups using non-abelian cohomology, where the group actions are not commutative, adding another layer of complexity to the theory.

History and relation to other fields

Group cohomology grew from ideas studied long before it was formally named in 1943–45. One early idea came from a theorem by David Hilbert in 1897, later linked to work in Galois theory. Other mathematicians explored similar concepts in the early 1900s when studying group extensions.

In the 1940s, several mathematicians discovered group cohomology independently. They included Samuel Eilenberg and Saunders Mac Lane in the United States, Heinz Hopf and Beno Eckmann in Switzerland, Hans Freudenthal in the Netherlands, and Dmitry Faddeev in the Soviet Union. Their work laid the foundation for using group cohomology in many areas, including algebraic number theory and physics.