Let \(X\) and \(Y\) be topological spaces, \(I = [0,1]\), and \(f, g\colon X\to Y\) continuous maps. The maps \(f\) and \(g\) are homotopic, written \(f\homotopic g\), if there is a continuous map \(H\colon X\cross I \to Y\) with \(H(x, 0) = f(x)\) and \(H(x, 1) = g(x)\) for all \(x\in X\); such a map \(H\) is a homotopy from \(f\) to \(g\). Two paths \(\gamma, \eta\colon I\to X\) with \(\gamma(0) = \eta(0)\) and \(\gamma(1) = \eta(1)\) are homotopic rel endpoints if there is a continuous map \(H\colon I\cross I\to X\) with \(H(s, 0) = \gamma(s)\), \(H(s, 1) = \eta(s)\), \(H(0,t) = \gamma(0)\), and \(H(1,t) = \gamma(1)\) for all \(s, t\in I\).
Homotopy is an equivalence relation on the set of continuous maps \(X\to Y\), and homotopy rel endpoints is an equivalence relation on the set of paths in \(X\) with given endpoints. The set of homotopy classes of continuous maps \(X\to Y\) is denoted \([X,Y]\).
The fundamental group \(\pi_1(X, x_0)\) is the set of classes of loops at \(x_0\) under homotopy rel endpoints.
[@Hat02]; [@Mun00].