Homology (mathematics)

Mathematically, homology is a useful method for studying shapes and spaces. It helps us understand key features of objects, such as how many holes a surface has or the loops in a graph. Homology began in a field of math called algebraic topology, where it changes geometric ideas into numbers and groups that can be worked with.

Homology examines sequences of special mathematical structures named abelian groups. These sequences, called chain complexes, have patterns that can be measured. By looking at these patterns, mathematicians form what are known as homology groups. These groups work like fingerprints for the original object, giving us important clues about its shape.

Applying homology to various mathematical objects gives us homology theories. These theories assist us in linking abstract ideas to real geometric spaces. For many usual spaces, different homology theories show the same outcomes, so we can discuss the homology of a topological space as one clear idea. This makes homology a helpful concept in popular mathematics, letting both experts and interested learners investigate the hidden structure of shapes, from the holes of a surface to the cycles of a graph.

Closely linked to homology is a concept named cohomology, which looks at similar patterns in another way using cochain complexes. Together, homology and cohomology supply mathematicians with strong tools for understanding the world of shapes and spaces.

Homology of chain complexes

In mathematics, a chain complex is a sequence of groups connected by special maps. These groups are called chains, and the maps show how they connect to each other. The important rule 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 are elements that map to zero. The group of boundaries are elements that come from the next level. The homology group at each level is formed by taking the cycles and removing the boundaries. This helps us understand important properties of the original sequence.

Homology theories

A homology theory is a way to study math problems. It changes the problems into something called chain complexes. Then we look at the homology of these complexes.

There are many types of homology theories. These include singular homology, Morse homology, Khovanov homology, and Hochschild homology. Each type uses a different way to make chain complexes. But they all try to get the same answers for the same problem.

In more advanced math, homology theories are special maps between groups. They help us understand how math objects are built. They can even show us when two objects are different.

Homology of a topological space

The term homology is 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.

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 helps describe its shape using groups of numbers. These groups are called homology groups. They tell us about the "holes" in the space.

For example, a circle has one one-dimensional hole. We can see this in its homology groups.

We can also look at simple shapes like spheres and tori. Each shape has different homology groups. These groups show how many holes the shape has and their size. This helps mathematicians understand the basic properties of spaces without studying 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.

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.

Types of homology

Main article: Simplicial homology

Homology helps mathematicians study shapes. One important type is simplicial homology. This looks at shapes made of triangles and similar pieces. By counting these pieces and how they fit together, mathematicians can find empty spaces in the shape.

Another type is singular homology, which works for any shape. It uses special maps to study the shape’s structure. There are many other homology theories for different math 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 sequences of groups 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 are important numbers we can calculate from these sequences. 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, we study sequences of groups called chain complexes. We can calculate something called the Euler characteristic. This helps us understand important features of what 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 useful tool in mathematics and science. It helps mathematicians prove important theorems, like the Brouwer fixed point theorem. This theorem 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.

In science and engineering, homology helps us understand sensor networks by looking at their structure. It is also used in physics to study how things change over time. It can help solve difficult equations in electromagnetic simulations by looking at the shape of the space where these equations work.

Software

Different programs can help us find homology groups for certain shapes. Linbox is a tool written in C++ that helps with quick math calculations. It 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 make problems easier before doing more complex math.

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

Some non-homology-based discussions of surfaces

Homology theory started with ideas about shapes and how we describe them. An important idea is the Euler polyhedron formula. This formula connects numbers about the corners, edges, and faces of solid shapes. Later, mathematicians looked at special numbers that describe surfaces. These surfaces include things like a sphere or a donut.

We can study simple shapes to see how homology works. On a sphere, any loop can be made smaller until it becomes a point. But on a torus (which looks like a donut), some loops cannot be made into points. For example, going around the hole of the torus or around the middle of it creates loops that stay big, no matter how you stretch the shape. These different loops help us understand the shape better.

When we cut and rearrange these loops, we can make new surfaces. For instance, cutting a torus along two loops and moving the pieces can create different shapes, like a Klein bottle. These new shapes have their own special features. Studying these loops and how they act helps mathematicians sort and understand different surfaces.

Main article: Analysis situs