  We prove an estimate for the difference of two solutions of the Schr\"odinger
map equation for maps from $T^1$ to $S^2.$ This estimate yields some continuity
properties of the flow map for the topology of $L^2(T^1,S^2)$, provided one
takes its quotient by the continuous group action of $T^1$ given by
translations. We also prove that without taking this quotient, for any $t>0$
the flow map at time $t$ is discontinuous as a map from
$\mathcal{C}^\infty(T^1,S^2)$, equipped with the weak topology of $H^{1/2},$ to
the space of distributions $(\mathcal{C}^\infty(T^1,\R^3))^*.$
