pi1

pi1, or the fundamental group, is a basic invariant in algebraic topology associated with a topological space X and a chosen basepoint x0 in X. It is defined as the set of homotopy classes of loops based at x0, where a loop is a continuous map γ: [0,1] → X with γ(0) = γ(1) = x0, and two loops are considered equivalent if they can be deformed into one another while keeping the endpoints fixed. The group operation is given by concatenation: the class of γ is multiplied by the class of δ by first traversing γ and then δ. The identity element is the class of the constant loop at x0, and the inverse of a loop γ is the loop γ reversed.

For spaces that are path-connected, the particular basepoint does not change the isomorphism class of the fundamental group: different basepoints yield isomorphic groups via a path connecting the basepoints. Examples illustrate common groups: π1(S^1) ≅ Z, reflecting loops winding around the circle; π1(S^2) is trivial; π1(T^2) ≅ Z × Z; π1(RP^2) ≅ Z/2.

Significance: π1(X,x0) encodes information about the space’s loop structure and is a cornerstone in topology. It classifies connected covering spaces of X (up to equivalence) and interacts with other areas of mathematics, including geometry and group theory. The concept is extended by the fundamental groupoid to handle multiple basepoints coherently.

Relation to higher theory: π1 may be non-abelian in general, unlike the higher homotopy groups πn(X) for n ≥ 2, which are abelian. The fundamental group also connects to the universal cover: the universal covering space X̃ of X has a free action of π1(X,x0) by deck transformations, and X ≅ X̃/π1(X,x0) in suitable settings.