Fundamental group

In mathematics, the fundamental group is one of the basic concepts of algebraic topology. Associated with every point of a topological space there is a fundamental group that conveys information about the 1-dimensional structure of the portion of the space surrounding the given point. The fundamental group is the first homotopy group.

Intuition and definition

Before giving a precise definition of the fundamental group, we try to describe the general idea in non-mathematical terms. Take some space, and some point in it, and consider all the loops at this point -- paths which start at this point, wander around as much they like and eventually return to the starting point. Two loops can be combined together in an obvious way: travel along the first loop, then along the second. The set of all the loops with this method of combining them is the fundamental group, except that for technical reasons it is necessary to consider two loops to be the same if one can be deformed into the other without breaking.

For the precise definition, let X be a topological space, and let x₀ be a point of X. We are interested in the set of continuous functions f : → X with the property that f(0) = x₀ = f(1). These functions are called loops with base point x₀. Any two such loops, say f and g, are considered equivalent if there is a continuous function h : ^*" target="_blank" >→ X with the property that, for all t in equivalence classes are called homotopy classes. The product f ∗ g of two loops f and g is defined by setting (f ∗ g)(t) = f(2t) if t is in ^*" target="_blank" >and (f ∗ g)(t) = g(2t − 1) if t is in g with twice the speed. The product of two homotopy classes of loops ^*" target="_blank" >and [g is then defined as ∗ g, and it can be shown that this product does not depend on the choice of representatives. With this product, the set of all homotopy classes of loops with base point x₀ forms the fundamental group of X at the point x₀ and is denoted π₁(X,x₀), or simply π(X,x₀). The identity element is the constant map at the basepoint, and the inverse of a loop f is the loop g defined by g(t) = f(1 − t). That is, g follows f backwards.

Although the fundamental group in general depends on the choice of base point, it turns out that, up to isomorphism, this choice makes no difference if the space X is path-connected. For path-connected spaces, therefore, we can write π(X) instead of π(X,x₀) without ambiguity whenever we care about the isomorphy class only.

Examples

In many spaces, such as Rⁿ, or any convex subset of Rⁿ, there is only one homotopy class of loops, and the fundamental group is therefore trivial, i.e. ({0},+). A path-connected space with a trivial fundamental group is said to be simply connected.

A more interesting example is provided by the circle. It turns out that each homotopy class consists of all loops which wind around the circle a given number of times (which can be positive or negative, depending on the direction of winding). The product of a loop which winds around m times and another that winds around n times is a loop which winds around m + n times. So the fundamental group of the circle is isomorphic to $(\mathbb{Z}\ , +)$ , the additive group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk–Ulam theorem in dimension 2.

Since the fundamental group is a homotopy invariant, the theory of the winding number for the complex plane minus one point is the same as for the circle.

Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be Abelian. For example, the fundamental group of a graph G is a free group. Here the rank of the free group is equal to 1 − χ(G): one minus the Euler characteristic of G. A somewhat more sophisticated example of a space with a non-Abelian fundamental group is the complement of a trefoil knot in R³.

Functoriality

If f : X → Y is a continuous map, x₀∈X and y₀∈Y with f(x₀) = y₀, then every loop in X with base point x₀ can be composed with f to yield a loop in Y with base point y₀. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π(X,x₀) to π(Y,y₀). This homomorphism is written as π(f) or f_*. We thus obtain a functor from the category of topological spaces with base point to the category of groups.

It turns out that this functor cannot distinguish maps which are homotopic relative the base point: if f and g : X → Y are continuous maps with f(x₀) = g(x₀) = y₀, and f and g are homotopic relative to {x₀}, then f_* = g_*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups.

The fundamental group functor takes products to products and coproducts to coproducts. That is, if X and Y are path connected, then π₁(X×Y)=π₁(X)×π₁(Y) and π₁(X∧Y)=π₁(X)*π₁(Y) where the wedge sum of topological spaces and the free product of groups are denoted in the latter formula. Both formulas generalize to arbitrary products. Furthermore the latter formula is a special case of the Seifert–van Kampen theorem which states that the fundamental group functor takes pushouts along inclusions to pushouts.

Relationship to first homology group

The fundamental groups of a topological space X are related to its first singular homology group, because a loop is also a singular 1-cycle. Mapping the homotopy class of each loop at a base point x₀ to the homology class of the loop gives a homomorphism from the fundamental group π(X,x₀) to the homology group H₁(X). If X is path-connected, then this homomorphism is surjective and its kernel is the commutator subgroup of π(X,x₀), and H₁(X) is therefore isomorphic to the abelianization of π(X,x₀). This is a special case of the Hurewicz theorem of algebraic topology.

Related concepts

The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used. The elements of the n-th homotopy group of X are homotopy classes of (basepointed) maps from Sⁿ to X.

The set of loops at a particular base point can be studied without regarding homotopic loops as equivalent. This larger object is the loop space.

Fundamental groupoid

Rather than singling out one point and considering the loops based at that point up to homotopy, one can also consider all paths in the space up to homotopy (fixing the initial and final point). This yields not a group but a groupoid, the fundamental groupoid of the space.