Homotopy group

In mathematics, homotopy groups are special tools used in algebraic topology to study and describe the shapes of different spaces. These groups help us understand how spaces are connected and what kinds of "holes" they might have. The simplest and most important of these groups is called the fundamental group, written as π₁(X). It looks at loops you can draw in a space and tells us how those loops can be stretched or shrunk into each other.

To understand higher homotopy groups, we look at maps of spheres into the space. For example, the nth homotopy group looks at how an n-dimensional sphere can be placed inside a space, again focusing on how these maps can be changed smoothly into one another. These collections of maps form a group, called the nth homotopy group, written as πₙ(X).

These groups are important because they give us strong information about the space. If two spaces have different homotopy groups, they cannot be the same shape in a topological sense. Even if two spaces are not exactly the same, they might still share the same homotopy groups, showing that they have similar shape properties.

The idea of homotopy, or how paths can be stretched into each other, was first introduced by the mathematician Camille Jordan.

Introduction

In mathematics, we often study complex ideas by linking them to simpler ones that keep important information. Homotopy groups are a way to connect groups — a type of mathematical structure — to topological spaces, which are shapes that can stretch and bend.

This connection helps mathematicians use ideas from group theory to understand topology. For example, two shapes might look different, like a torus (which has a "hole") and a sphere (which doesn’t), and homotopy groups can show these differences clearly. Even though it’s hard to see these global differences just by looking locally, homotopy groups capture the overall shape.

Definition

In math, we study shapes and how they can stretch and bend without tearing. One way to understand shapes is by looking at loops and paths on them.

The simplest idea is called the fundamental group fundamental group. It looks at loops that start and end at the same point, and how these loops can be combined.

For more complex shapes, we use homotopy groups. These help us understand more about the shape by looking at maps from spheres or cubes. When we combine these maps in a special way, we get a group — a collection of rules for how these maps can work together.

For most shapes, choosing different starting points doesn’t change the basic information we get, except in some special cases. This helps us study the overall shape more easily.

Long exact sequence of a fibration

When we study shapes in mathematics, we often look at how loops and paths behave in spaces. One important tool is the "long exact sequence of a fibration," which helps us understand the relationships between different spaces.

Imagine three spaces connected in a special way: a larger space, a smaller space inside it, and a base space. The long exact sequence tells us how the loops in these spaces relate to each other. For example, in the case of the Hopf fibration, we can see that certain loops in a 3-dimensional sphere are the same as loops in a 2-dimensional sphere.

This idea is useful in many areas of mathematics, helping us understand the shapes of spaces better.

Methods of calculation

Finding homotopy groups can be very hard, much harder than finding some other important shapes in math. There isn’t a simple way to break a space into smaller pieces to find its homotopy groups, unlike for the fundamental group. But new ideas from the 1980s helped scientists learn more about these groups.

For some special shapes, like tori, the higher homotopy groups are very simple. However, even for basic shapes like spheres, we still don’t know all their homotopy groups. To find these groups, scientists need very advanced tools. Some groups can be found by comparing them to other shape groups using a special rule.

A list of methods for calculating homotopy groups

Here are some important methods used to find homotopy groups:

• The long exact sequence of homotopy groups of a fibration.
• The Hurewicz theorem, which comes in different forms.
• The Blakers–Massey theorem, also called excision for homotopy groups.
• The Freudenthal suspension theorem, which is a result related to excision for homotopy groups.

Relative homotopy groups

Homotopy groups can also be studied for pairs of spaces. This means we look at a space (X) and a part of it called (A). We can study how loops and shapes behave when part of them must stay inside (A).

These special groups are called relative homotopy groups, written as (\pi_n(X, A)). They help us understand the shape of (X) while considering how it connects to the part (A). When (A) is just a single point, these groups match the usual homotopy groups of the space (X).

Related notions

Homotopy groups are important in a part of math called homotopy theory, which helped create another idea called model categories. We can also talk about homotopy groups in a special kind of math shape called simplicial sets.

Homology groups are like homotopy groups because they can both show us the "holes" in a shape. But homotopy groups can be really tricky to figure out. Homology groups, on the other hand, are simpler because they follow a certain rule, and so do higher homotopy groups. For any shape X, its nth homotopy group is written as <sup>π</sup><sub>n</sub>(X), and its nth homology group is written as H<sub>n</sub>(X) or H<sub>n</sub>(X; <sup>Z</sup>).

Main article: homotopy theory

Main articles: Homology (mathematics)