Homology (mathematics)

In mathematics, homology is a powerful tool used to study shapes and spaces. It helps us understand important features of objects, like the number of holes in a surface or the loops in a graph. Homology was first developed in a branch of math called algebraic topology, where it helps turn geometric ideas into numbers and groups that can be calculated.

Homology works by looking at sequences of special mathematical structures called abelian groups. These sequences, known as chain complexes, have patterns that can be measured. By studying these patterns, mathematicians create what are called homology groups. These groups act like fingerprints for the original object, telling us deep information about its shape.

When we apply homology to different kinds of mathematical objects, we get homology theories. These theories help us connect abstract ideas to real geometric spaces. For many common spaces, different homology theories give the same results, allowing us to talk about the homology of a topological space as if it were a single, clear idea. This makes homology a valuable concept in popular mathematics, helping both experts and curious learners explore the hidden structure of shapes, from the holes of a surface to the cycles of a graph.

Closely related to homology is a concept called cohomology, which studies similar patterns in a slightly different way using cochain complexes. Together, homology and cohomology give mathematicians strong tools for understanding the world of shapes and spaces.

Homology of chain complexes

In mathematics, a chain complex is like a sequence of groups connected by special maps. These groups are called chains, and the maps show how they connect to each other. The key idea is that applying two maps in a row always gives zero.

From this setup, we can create two important groups for each level: the group of cycles, which are elements that map to zero, and the group of boundaries, which are elements that come from the next level. The homology group at each level is formed by taking the cycles and "modding out" the boundaries, which helps us understand important properties of the original sequence.

Homology theories

A homology theory is a way to study mathematical objects by first turning them into chain complexes and then looking at the homology of these complexes. Different types of homology theories, like singular homology, Morse homology, Khovanov homology, and Hochschild homology, use different methods to create these chain complexes. The goal is that all these methods give the same results for the same object.

In more advanced math, homology theories are seen as special mappings between categories. They help us understand the structure of mathematical objects and can even tell us when two objects are different.

Homology of a topological space

The term homology is often used to study topological spaces. Homology helps us understand the shape of these spaces by looking at their "holes." For example, a circle has one hole, while a sphere has none. This idea comes from noticing that certain shapes can be told apart by counting these holes.

Homology uses special groups called homology groups to capture these properties. These groups are built from cycles (which represent loops or closed shapes) and boundaries (which represent parts of shapes that have edges). By studying these groups, we can learn important features about the space, like how many holes it has and of what sizes.

Informal examples

The homology of a topological space describes its shape using groups of numbers. These groups, called homology groups, tell us about the "holes" in the space. For example, a circle has one one-dimensional hole, which shows up in its homology groups.

We can see this with simple shapes like circles, spheres, and tori. Each shape has different homology groups that reflect how many holes it has and of what size. This helps mathematicians understand the basic properties of spaces without looking at all their details.

For the homology groups of a graph, see graph homology.

Construction of homology groups

The construction of homology groups starts with an object like a shape or space, called X. We create something called a chain complex C(X), which is a sequence of groups connected by special maps known as boundary operators. These operators help us understand how different parts of the space connect to each other.

From this chain complex, we build the _n_th homology group of X by looking at certain parts of the sequence and comparing them. These groups tell us important information about the original object X, such as how many "holes" it has in different dimensions.

Main article: simplicial homology

Homology vs. homotopy

The nth homotopy group π_n(X) of a space X is a way to study shapes by looking at loops and their properties. The most basic one is the fundamental group π₁(X). For connected spaces, the Hurewicz theorem connects homotopy groups to homology groups H_n(X) through a special mapping called the Hurewicz homomorphism.

When n is greater than 1, these groups can be tricky to understand. But for n = 1, the Hurewicz homomorphism helps us see that the first homology group H₁(X) is closely related to the fundamental group. For example, if X is a figure eight shape, its first homotopy group is not commutative, meaning the order of loops matters. However, its first homology group becomes abelian, meaning the order no longer matters, showing a key difference between homotopy and homology.

Types of homology

Main article: Simplicial homology

Homology is a way to study shapes in mathematics. One important type is simplicial homology, which looks at shapes made of triangles and their higher-dimensional versions, called simplices. By counting these pieces and how they connect, mathematicians can find "holes" in the shape — spaces that are not filled in.

Another type is singular homology, which works for any space, not just triangle-based ones. It uses continuous maps from simplices to study the shape’s structure. There are many other homology theories, each suited to different kinds of mathematical problems. These include Borel–Moore homology, Cellular homology, Cyclic homology, Hochschild homology, Floer homology, Intersection homology, K-homology, Khovanov homology, Morse homology, Persistent homology, and Steenrod homology.

Main article: Singular homology

Main article: Group cohomology

Homology functors

Chain complexes are like sequences of groups that are connected by special maps. We can think of these maps as arrows pointing from one group to another in the sequence. When we have two such sequences, we can create a way to match up the groups in one sequence with the groups in the other using special rules.

Homology groups, which are important numbers we can calculate from these sequences, behave in a special way. They act like "detectors" that tell us how the sequences are related. This idea helps us understand the structure of more complicated mathematical objects by looking at these simpler sequences.

The main difference between homology and cohomology is how these sequences depend on the object we are studying. In homology, they depend in one way, while in cohomology, they depend in the opposite way. This small difference leads to important uses in many areas of mathematics.

Properties

In mathematics, when we study sequences of groups called chain complexes, we can calculate something called the Euler characteristic. This helps us understand important features of the object we are studying.

When we have special sequences of these chain complexes, we can create longer sequences of homology groups. These longer sequences help us find and calculate homology groups more easily, using tools like the zig-zag lemma.

Applications

Homology is a powerful tool in mathematics and science. It helps mathematicians prove important theorems, like the Brouwer fixed point theorem, which says that any map of a ball to itself has a fixed point. It is also used in topological data analysis to study shapes in data sets by creating approximations and calculating their properties.

In science and engineering, homology helps understand sensor networks by looking at their structure, and it aids in physics by studying the behavior of dynamical systems. It is also useful in solving complex equations in electromagnetic simulations by considering the shape of the space in which they occur.

Software

Different programs have been made to help calculate homology groups for certain mathematical shapes. Linbox is a tool written in C++ that helps with quick math calculations and works with other programs like Gap and Maple. Other tools like Chomp, CAPD::Redhom, and Perseus are also written in C++ and use special methods to simplify problems before doing more complex math.

Another program, Kenzo, is written in Lisp and can also help study other math properties besides homology. Gmsh has a feature to solve homology problems for shapes used in engineering and can create special math setups that other engineering programs can use directly.

Some non-homology-based discussions of surfaces

Homology theory began with ideas about shapes and how we can describe them. One important starting point is the Euler polyhedron formula, which relates numbers about the corners, edges, and faces of solid shapes. Later mathematicians studied special numbers that describe how a surface, like a sphere or a donut, can be stretched or twisted without tearing.

We can look at simple shapes to see how homology works. On a sphere, any loop can be shrunk to a point. But on a torus (which looks like a donut), there are loops that cannot be shrunk to points. For example, going around the hole of the torus or around the middle of the torus creates loops that stay big no matter how you stretch the shape. These different kinds of loops help us understand the shape better.

When we cut and rearrange these loops, we can create new surfaces. For instance, cutting a torus along two loops and rearranging the pieces can make different shapes, like a Klein bottle. These new shapes have their own special properties, such as whether you can tell left from right when you walk along a loop. Studying these loops and how they behave helps mathematicians classify and understand different surfaces.

Main article: Analysis situs