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2 Examples 3 Functoriality 4 Relationship to first homology group 5 Related concepts |
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 base point 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 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 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 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.
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.
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 non-Abelian fundamental group is the complement of a trefoil knot in R3. 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.
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 {x0}, then f* = g*. As a consequence, two homotopy equivalent path-connected spaces have isomorphic fundamental groups.
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).
The fundamental group measures the 1-dimensional hole structure of a space. For studying "higher-dimensional holes", the homotopy groups are used.
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.Intuition and definition
Examples
Functoriality
Relationship to first homology group
Related concepts
Fundamental groupoid