Fundamental group

In mathematics, the fundamental group is one of the basic concepts of algebraic topology. It is a group associated with every point of a topological space and conveying information about the 1dimensional structure of the space. The fundamental group is the first homotopy group.
Contents 
Intuition and definition
Before giving a precise definition of the fundamental group, we try to describe the general idea in nonmathematical 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 x_{0} be a point of X. We are interested in the set of continuous functions f : [0,1] → X with the property that f(0) = x_{0} = f(1). These functions are called loops with base point x_{0}. Any two such loops, say f and g, are considered equivalent if there is a continuous function h : [0,1] × [0,1] → X with the property that, for all t in [0,1], h(t,0) = f(t), h(t,1) = g(t) and h(0,t) = x_{0} = h(1,t). Such an h is called a homotopy from f to g, and the corresponding 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 [0,1/2] and (f * g)(t) = g(2t1) if t is in [1/2,1]. The loop f * g thus first follows the loop f with "twice the speed" and then follows g with twice the speed. The product of two homotopy classes of loops [f] and [g] is then defined as [f * 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 x_{0} forms the fundamental group of X at the point x_{0} and is denoted π_{1}(X,x_{0}), or simply π(X,x_{0}). 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(1t). 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 pathconnected. For pathconnected spaces, therefore, we can write π(X) instead of π(X,x_{0}) without ambiguity whenever we care about the isomorphy class only.
Examples
In many spaces, such as R^{n}, or any convex subset of R^{n}, there is only one homotopy class of loops, and the fundamental group is therefore trivial. A pathconnected 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 Z, the group of integers. This fact can be used to give proofs of the Brouwer fixed point theorem and the BorsukUlam 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 many of the other groups associated with a topological space, the fundamental group need not be Abelian. An example of a space with a nonAbelian fundamental group is the complement of a trefoil knot in R^{3}. If several circles are joined together at a point, the fundamental group is a free group, with generators loops going round just one of the circles.
Functoriality
If f : X → Y is a continuous map, x_{0}∈X and y_{0}∈Y with f(x_{0}) = y_{0}, then every loop in X with base point x_{0} can be composed with f to yield a loop in Y with base point y_{0}. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π(X,x_{0}) to π(Y,y_{0}). 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(x_{0}) = g(x_{0}) = y_{0}, and f and g are homotopic relative to {x_{0}}, then f_{*} = g_{*}. As a consequence, two homotopy equivalent pathconnected spaces have isomorphic fundamental groups.
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 1cycle. Mapping the homotopy class of each loop at a base point x_{0} to the homology class of the loop gives a homomorphism from the fundamental group π(X,x_{0}) to the homology group H_{1}(X). If X is pathconnected, then this homomorphism is surjective and its kernel is the commutator subgroup of π(X,x_{0}), and H_{1}(X) is therefore isomorphic to the abelianization of π(X,x_{0}).
Related concepts
The fundamental group measures the 1dimensional hole structure of a space. For studying "higherdimensional holes", the homotopy groups are used. The elements of the nth homotopy group of X are homotopy classes of (basepointed) maps from S^{n} to X.
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.