Preliminary and the classical homotopy theory; Decompositions of self smash products; Decompositions of the loop spaces; The homotopy groups $\pi_{n+r}(\Sigma^n\mathbb{R}\mathrm{P}^2)$ for $n\geq 2$ and $r\leq8$; The homotopy theory of $\Sigma\mathbb{R}\mathrm{P}^2$; Bibliography.