III.39 Homotopy Groups

If X is a TOPOLOGICAL SPACE [III.90], then a loop in X is a path that begins and ends at the same point; or, more formally, a continuous function f :[0, 1] → X such that f(0)= f(1). The point where the path begins and ends is called the base point. If two loops have the same base point, they are called homotopic if one can be continuously deformed to the other, with all the intermediate paths living in X and beginning and ending at the given base point. For example, if X is the plane ², then any two paths that begin and end at (0, 0) are homotopic, whereas if X is the plane with the origin removed, then whether or not two paths (that begin and end at some other point) are homotopic depends on whether or not they go around the origin the same number of times.

Homotopy is an EQUIVALENCE RELATION [I.2 §2.3], and the equivalence classes of paths with base point x form the fundamental group of X, relative to x, which is denoted by π_l (X, x). If X is connected, then this does not depend on x and we can write π₁ (X) instead. The group operation is “concatenation”: given two paths that begin and end at x, their “product” is the combined path that goes along one and then the other, and the product of equivalence classes is then defined to be the equivalence class of the product. This group is a very important invariant (see for instance GEOMETRIC AND COMBINATORIAL GROUP THEORY [IV.10 §7]); it is the first in a sequence of higher-dimensional homotopy groups, which are described in ALGEBRAIC TOPOLOGY [IV.6 §§2, 3].