We undertake a study of topological properties of the real Mordell-Weil group $\operatorname{ MW}_{ \mathbb R}$ of real rational elliptic surfaces $X$ which we accompany by a related study of real lines on $X$ and on the "subordinate" del Pezzo surfaces $Y$ of degree 1. We give an explicit description of isotopy types of real lines on $Y_{ \mathbb R}$ and an explicit presentation of $\operatorname{ MW}_{ \mathbb R}$ in the mapping class group $\operatorname{ Mod}(X_{ \mathbb R})$. Combining these results we establish an explicit formula for the action of $\operatorname{ MW}_{ \mathbb R}$ in $H_1(X_{ \mathbb R})$.
Categories: math.AG math.GT