Fundamental Group - Definition

Definition

Let X be a topological space, and let x0 be a point of X. We are interested in the following set of continuous functions called loops with base point x0.

Now the fundamental group of X with base point x is this set modulo homotopy

equipped with the group multiplication defined by (fg)(t) := f(2t) if 0 ≤ t ≤ 1/2 and (fg)(t) := g(2t − 1) if 1/2 ≤ t ≤ 1. Thus the loop fg 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 and is then defined as, and it can be shown that this product does not depend on the choice of representatives.

With the above 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

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 (actually, even up to inner isomorphism), this choice makes no difference as long as the space X is path-connected. For path-connected spaces, therefore, we can write π1(X) instead of π1(X, x0) without ambiguity whenever we care about the isomorphism class only.

Read more about this topic:  Fundamental Group

Famous quotes containing the word definition:

    Although there is no universal agreement as to a definition of life, its biological manifestations are generally considered to be organization, metabolism, growth, irritability, adaptation, and reproduction.
    The Columbia Encyclopedia, Fifth Edition, the first sentence of the article on “life” (based on wording in the First Edition, 1935)

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)

    Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.
    Nadine Gordimer (b. 1923)