Let \(X\) and \(Y\) be topological spaces. The homotopy class \([f]\) of a continuous map \(f\colon X\to Y\) is its equivalence class under the relation \(\simeq\) of being homotopic. The set of homotopy classes of continuous maps \(X\to Y\) is \[ [X, Y]\coloneqq\Hom_{\Top}(X, Y)/\simeq. \]
If \(f, g\colon X\to Y\) are homotopic, then \(f_* = g_*\colon H_n(X)\to H_n(Y)\) for every \(n\).
For a based space \((Y, y_0)\) and \(n\geq 1\), the homotopy group \(\pi_n(Y, y_0)\) is the set \([(S^n, s_0), (Y, y_0)]\) of homotopy classes of based maps under homotopies fixing the basepoint.
See [@Hat02].