Let \(X\) be a topological space and \(x_0\in X\), and let \(I = [0,1]\). The fundamental group \(\pi_1(X, x_0)\) is the set of classes of loops \(\gamma\colon I\to X\) based at \(x_0\) under homotopy rel endpoints, with multiplication \[ [\gamma][\eta]\coloneqq[\gamma\cdot\eta], \] where \(\gamma\cdot\eta\) is the concatenation of \(\gamma\) and \(\eta\).
The multiplication on \(\pi_1(X, x_0)\) is well defined and makes \(\pi_1(X, x_0)\) a group. Its identity is the class of the constant loop at \(x_0\), and \(\inverseof{[\gamma]} = [\bar\gamma]\), where \(\bar\gamma(s)\coloneqq\gamma(1-s)\).
Let \(h\colon I\to X\) be a path from \(x_0\) to \(x_1\), with reverse \(\bar h\). Then \(\beta_h\colon\pi_1(X, x_1)\to\pi_1(X, x_0)\), \(\beta_h[\gamma]\coloneqq[h\cdot\gamma\cdot\bar h]\), is a group isomorphism. It depends only on the homotopy class of \(h\) rel endpoints, and for a second path \(h'\) from \(x_0\) to \(x_1\), \(\beta_{h'} = c\circ\beta_h\), where \(c\) is conjugation by \([h'\cdot\bar h]\in\pi_1(X, x_0)\). In particular, if \(X\) is path connected, the groups \(\pi_1(X, x)\) for \(x\in X\) are all isomorphic.
See [@Hat02].