Let \((X,x_0)\) be a based space and \(s_0\in S^1\) a basepoint. The loop space of \(X\) is the set \[ \Omega X\coloneqq\ts{\gamma\colon S^1\to X \st \gamma\text{ continuous},\ \gamma(s_0)=x_0} \] with the compact-open topology, based at the constant loop at \(x_0\) [@Hat02].
Let \((X,x_0)\) and \((Y,y_0)\) be based spaces, and write \([-,-]_*\) for based homotopy classes of based maps and \(\Sigma\) for reduced suspension.
\(\pi_n(\Omega X)\cong\pi_{n+1}(X,x_0)\) for all \(n\geq0\).
There is a bijection \([\Sigma X,Y]_*\cong[X,\Omega Y]_*\), natural in \(X\) and \(Y\).
[@Hat02]