Cohomology

In mathematics, especially in areas like homology theory and algebraic topology, cohomology is a useful idea. It helps mathematicians study shapes and spaces. Cohomology gives these spaces special algebraic structures. It provides more detailed information about a space than other methods, like homology.

Cohomology starts from homology but uses a different way of thinking. Instead of looking at the space directly, it looks at functions on the space. These functions help show important features of the space. This makes it useful for solving many problems.

Over time, cohomology has become very important in many parts of mathematics. It connects to geometry and algebra. It appears in many different areas of advanced math. One key feature of cohomology is a special "multiplication" rule called the cup product. This rule adds extra structure, making cohomology even more useful than homology alone.

Singular cohomology

Singular cohomology is a way to study shapes in mathematics. It helps us learn about spaces by giving them special numbers or groups. These groups tell us more than simpler methods.

This method starts with chains, which are like building blocks from maps of simple shapes into the space. By looking at these chains in a special way, we can create 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 useful 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.

Poincaré duality

Main article: Poincaré duality

Poincaré duality is a big idea in mathematics. It helps us learn about the shape of certain spaces.

For a special kind of space, the cohomology groups—which are like math descriptions of the space—have a special relationship.

If the space has a dimension of n, one of its cohomology groups is very simple. The other groups come in pairs that match exactly in size and structure. This means learning about one group tells us everything about its partner group. This idea is very helpful for studying the shape and features of complicated spaces.

Characteristic classes

Main article: Characteristic class

In mathematics, a characteristic class is a special tool that helps describe properties of vector bundles. Vector bundles are ways of assigning vectors to each point in space.

One important example is the Euler class. It 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 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 space used to study cohomology. For each mathematical group and a number, there is a space where only one homotopy group is not zero. This helps us understand cohomology, which is a way to give spaces algebraic information.

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 part of this 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 connect two math ideas about a space. It links two types of math descriptions of the space, named homology and cohomology. This helps us understand the space better by using algebra.

When we use the cap product on special parts of a space, called submanifolds, we can discover new parts by seeing how these special parts meet. This gives us helpful ideas about the shape and design 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 tool in math that helps us study shapes using algebra. It gives us better ways to look at spaces. In the 1950s, mathematicians started using sheaf cohomology more often to understand shapes and complicated functions.

A mathematician named Grothendieck helped make this idea clearer. He showed how sheaf cohomology links to other algebra tools. It helps us learn about many math structures by studying 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.

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.

Axioms and generalized cohomology theories

See also: List of cohomology theories

There are different ways to find out about shapes in topology. Sometimes these ways give different answers, but they often agree on 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. These rules include giving the same answer for shapes that can be smoothly changed 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 help us study many kinds of mathematical structures, not just spaces. These include groups, rings, and geometric shapes.

Some examples are Algebraic K-theory, Čech cohomology, Coherent sheaf cohomology, Group cohomology, and Quantum cohomology. There are many more, each helping mathematicians learn about different properties of shapes and structures.