Cohomology

In mathematics, especially in areas like homology theory and algebraic topology, cohomology is a powerful tool. It helps mathematicians study shapes and spaces by assigning them special algebraic structures. Think of it as a way to give more detailed information about a space than other methods, like homology.

Cohomology often starts from homology but uses a different approach. Instead of looking at the space directly, it looks at functions on the space. These functions help capture important features of the space in a way that is useful for many problems.

Over time, cohomology has become very important in many parts of mathematics. It connects to geometry and algebra, and it shows up in many different areas of advanced math. One key feature of cohomology is that it often has a special "multiplication" rule called the cup product, which gives it an extra layer of structure, making it even more useful than homology alone.

Singular cohomology

Singular cohomology is a way to study shapes in mathematics. It helps us understand the properties of spaces by assigning them special numbers or groups. These groups give us more information than simpler methods.

This method starts with chains, which are like building blocks made from maps of simple shapes into the space. By looking at these chains in a special way, we can create what’s called cohomology groups. These groups help us see deeper qualities of the space, like how it is connected or shaped. One special feature of cohomology is the cup product, which lets us combine these groups in a meaningful way. This helps mathematicians solve many problems about shapes and their properties.

Examples

Cohomology helps us understand the shape of spaces using algebra. For a single point or any space that can be "shrunk" to a point, the cohomology is very simple: it’s just the whole numbers (called Z) in degree 0.

For more interesting shapes, like spheres, tori (the shape of a donut), and projective spaces, cohomology gives richer algebraic structures. For example, the cohomology of a sphere looks like a polynomial ring where one special element has a square equal to zero. The cohomology of a torus is like an "exterior algebra," which means it behaves in a specific way when you multiply its generators.

These examples show how cohomology can capture important features of a space using algebraic objects.

The diagonal

The cup product in cohomology comes from something called the diagonal map. This map takes a space and pairs each point with itself.

For any two spaces, we can create a special product of their cohomology classes. This product can also be described using the cup product and projections from the combined space to each original space.

Poincaré duality

Main article: Poincaré duality

Poincaré duality is a concept in mathematics that helps us understand the structure of certain spaces. It tells us that for a special kind of space called a closed oriented manifold, the cohomology groups — which are like algebraic descriptions of the space — have a strong relationship with each other.

Specifically, if we have a space of dimension n, one of its cohomology groups is very simple, and the other groups come in pairs that match up perfectly in size and structure. This means that studying one group gives us complete information about its partner group. This idea is very useful in algebraic topology for understanding the shape and properties of complex spaces.

Characteristic classes

Main article: Characteristic class

In mathematics, a special kind of object called a characteristic class helps describe properties of vector bundles, which are ways of assigning vectors to each point in space. One important example is the Euler class, which gives information about how a vector bundle behaves over a space.

There are other characteristic classes too, such as Chern classes, Stiefel–Whitney classes, and Pontryagin classes. These all provide different ways to understand and work with vector bundles using cohomology.

Eilenberg–MacLane spaces

Main article: Eilenberg–MacLane space

In mathematics, an Eilenberg–MacLane space is a special kind of space used to study cohomology. For each mathematical group and a natural number, there is a space where only one homotopy group is non-zero. This makes it a useful tool for understanding cohomology, which is a way to assign algebraic information to spaces.

For example, the space K(ℤ, 1) can be the circle S¹. This helps us understand the first cohomology group with integer coefficients. It shows that every element of this cohomology group comes from a point on the circle through a map from the space to the circle.

Cap product

Main article: Cap product

The cap product is a way to combine two algebraic objects related to a space. It helps connect two types of mathematical descriptions of the space, called homology and cohomology. This connection allows us to understand the space better by using algebra.

When we use the cap product on certain special parts of a space, called submanifolds, we can find new parts by looking at how these special parts intersect. This gives us useful information about the shape and structure of the space.

Brief history of singular cohomology

Cohomology is a big idea in mathematics that helps us understand shapes better. For many years after homology — a similar idea — was created, people didn’t fully see how important cohomology would become. The idea of cohomology started with Henri Poincaré when he worked on something called Poincaré duality.

Later, mathematicians like J. W. Alexander and Solomon Lefschetz worked on how to measure when different parts of a shape meet each other. In the 1930s, Georges de Rham showed a connection between homology and special mathematical objects called differential forms. Over time, many mathematicians such as Heinz Hopf, Egbert van Kampen, Lev Pontryagin, Norman Steenrod, Hassler Whitney, and Eduard Čech added more pieces to the puzzle. Finally, in the 1940s, Samuel Eilenberg gave us the modern way to define cohomology, and together with Steenrod, they created rules to understand these theories better.

Sheaf cohomology

Main article: Sheaf cohomology

Sheaf cohomology is a way to study shapes in mathematics using ideas from algebra. It lets us look at spaces with more detailed tools than before. In the 1950s, mathematicians began using sheaf cohomology a lot in studying shapes and complex functions.

This idea was made clearer by a mathematician named Grothendieck. He showed how sheaf cohomology connects to other algebraic tools. It helps us understand many different kinds of mathematical structures by looking at their "right derived functors."

Cohomology of varieties

There are many ways to calculate the cohomology of algebraic varieties. For simple cases, like smooth projective varieties over a field of characteristic 0, tools from Hodge theory, known as Hodge structures, help find the cohomology. In the easiest case, the cohomology of a smooth hypersurface can be worked out just from the degree of a special math rule called a polynomial.

When dealing with varieties over a finite field, or fields of characteristic p, stronger tools are needed because normal homology and cohomology methods don’t work well. This is because these varieties end up being just a finite set of points. A mathematician named Grothendieck created something called a Grothendieck topology and used sheaf cohomology with the étale topology to build a new cohomology theory for these varieties. This leads to what is called ℓ-adic cohomology, which is a special kind of cohomology that works even when the field has a certain property called a finite characteristic.

Axioms and generalized cohomology theories

See also: List of cohomology theories

There are different ways to define cohomology for shapes in topology. These methods sometimes give different answers, but they agree on many important shapes. One way to understand this is through a list of rules called the Eilenberg–Steenrod axioms. Any method that follows these rules will give the same results for certain kinds of spaces.

In the 1960s, a mathematician named George W. Whitehead suggested ignoring one of these rules. This led to the idea of "generalized cohomology theories." These theories, like K-theory, can give extra information about a space that regular cohomology cannot.

A generalized cohomology theory follows certain rules, such as giving the same answer for shapes that can be continuously deformed into each other, creating sequences of related groups, and working well with combinations of spaces. These theories help mathematicians study spaces in deeper ways.

Other cohomology theories

Cohomology theories can be used to study many different kinds of mathematical structures, not just spaces. These include algebraic structures like groups and rings, as well as geometric shapes and more.

Some examples of these broader cohomology theories are Algebraic K-theory, Čech cohomology, Coherent sheaf cohomology, Group cohomology, and Quantum cohomology. There are many others, each helping mathematicians understand different properties of shapes and structures.