Homotopy Group

Introduction

In algebraic topology, a branch of mathematics, the concept of a homotopy group is a fundamental tool used to classify topological spaces. Homotopy groups are algebraic invariants that encode information about the shape and structure of spaces. They are particularly useful in distinguishing spaces that are not homotopy equivalent, meaning they cannot be continuously deformed into each other.

Definition and Basic Properties

Homotopy groups are defined using the notion of homotopy, which is a continuous deformation between two continuous functions. Specifically, the n-th homotopy group of a pointed space \((X, x_0)\), denoted \(\pi_n(X, x_0)\), is the set of homotopy classes of maps from the n-dimensional sphere \(S^n\) to \(X\) that preserve the base point \(x_0\). Formally, \(\pi_n(X, x_0) = [S^n, X]_{x_0}\).

For \(n = 0\), the homotopy group \(\pi_0(X)\) is the set of path-connected components of \(X\). For \(n \geq 1\), \(\pi_n(X, x_0)\) is a group, and for \(n \geq 2\), it is an abelian group. The group operation is induced by the concatenation of loops.

Fundamental Group

The first homotopy group, \(\pi_1(X, x_0)\), is known as the fundamental group. It captures information about the loops in \(X\) based at \(x_0\) and their homotopy classes. The fundamental group is a powerful invariant in topology, providing insights into the structure and properties of the space.

The fundamental group can be computed for various spaces. For instance, the fundamental group of the circle \(S^1\) is isomorphic to the integers \(\mathbb{Z}\), reflecting the fact that loops around the circle can be wound any number of times in either direction.

Higher Homotopy Groups

For \(n \geq 2\), the higher homotopy groups \(\pi_n(X, x_0)\) provide deeper insights into the topology of the space. These groups are abelian and can be studied using various algebraic techniques. Higher homotopy groups are particularly important in the study of spheres, complexes, and other higher-dimensional topological spaces.

For example, the higher homotopy groups of spheres are of significant interest. The homotopy groups of the n-sphere \(S^n\) are known to be trivial for \(k < n\) and non-trivial for \(k = n\). For \(k > n\), the groups \(\pi_k(S^n)\) can be highly non-trivial and are the subject of ongoing research in algebraic topology.

Homotopy Groups of Spheres

The homotopy groups of spheres are a central topic in algebraic topology. The study of these groups has led to the development of many important concepts and techniques. For instance, the Hopf fibration is a famous example that provides a non-trivial element in \(\pi_3(S^2)\).

The computation of homotopy groups of spheres is a challenging problem. While some groups are well-understood, many remain mysterious. For example, \(\pi_3(S^2) \cong \mathbb{Z}\) and \(\pi_4(S^3) \cong \mathbb{Z}/2\mathbb{Z}\), but the structure of higher homotopy groups becomes increasingly complex.

Tools and Techniques

Several tools and techniques have been developed to study homotopy groups. These include:

**Homotopy exact sequence**: This sequence relates the homotopy groups of a space, a subspace, and their quotient.
**Hurewicz theorem**: This theorem connects homotopy groups with homology groups, providing a bridge between algebraic topology and homological algebra.
**Whitehead theorem**: This theorem gives conditions under which a map that induces isomorphisms on homotopy groups is a homotopy equivalence.

Applications

Homotopy groups have numerous applications in mathematics and beyond. They are used in the classification of fiber bundles, the study of manifolds, and in homotopy theory itself. In physics, homotopy groups play a role in the study of topological defects and gauge theory.

References