Fundamental group

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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 1-dimensional structure of the space. The fundamental group is the first homotopy group.

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 : [0,1] -> X with the property that f(0) = x0 = f(1). These functions are called loops with basepoint x0. 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) = x0 = 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(2t-1) if t is in [1/2,1]. 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 basepoint x0 forms the fundamental group of X at the point x0 and is denoted π1(X,x0), or simply π(X,x0).

Although the fundamental group in general depends on the choice of basepoint, it turns out that, upto 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.

In many spaces, such as Rn, there is only one homotopy class of loops, and the fundamental group is therefore trivial. 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 Z, the group of integers.

Unlike many of the other groups associated with a topological space, the fundamental group need not be Abelian. In fact, every group is isomorphic to the fundamental group of some topological space. An example of a space with a non-Abelian fundamental group is the complement of a trefoil knot in R3.