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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). 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.
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. 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 integersThe integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3,. and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which st. This fact can be used to give proofs of the Brouwer fixed point theoremTopology Theorems In mathematics, the Brouwer fixed point theorem states that every continuous function from the closed unit ball D ''n to itself has a fixed point. In this theorem, n is any positive integer, and the closed unit ball is the set of all poi and the Borsuk-Ulam theoremAlgebraic topology Theorems The Borsuk-Ulam theorem states that any continuous function from an n sphere into Euclidean n space maps some pair of antipodal points to the same point. Two points on a sphere are called antipodal if they sit on directly oppos in dimension 2.
Since the fundamental group is a homotopy invariant, the theory of the winding numberAlgebraic topology Complex analysis In mathematics, the winding number is a topological invariant playing a leading role in complex analysis. Intuitively, the winding number of a curve γ with respect to a point z is the number of times γ goes 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 AbelianAbstract algebra Algebra Group theory In mathematics, an abelian group is a commutative group, i. a group G ) such that a b b a for all a and b in G''. Abelian groups are named after Niels Henrik Abel. Notation There are two main notational conventions fo. An example of a space with a non-Abelian fundamental group is the complement of a trefoil knotIn knot theory, the trefoil knot is the simplest nontrivial knot. It can be obtained by joining the loose ends of an overhand knot. It is the unique prime knot with three crossings. It can be described as a (2,3 torus knot, its braid word being σ3. in R3. If several circles are joined together at a point, the fundamental group is a free groupIn mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses (disregarding trivial variations such as st-1 su-1ut-1 ., with generators loops going round just one of the circles.
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