  In the following short note, we give a new geometric interpretation of the
generator of the infinite cyclic group
$H^1(\text{Mod}(S_{g,1});H^1(S_g;\mathbb{Z}))$ (this computation is proved by
Morita). There are several construction of this class given by Earle, Morita,
Trapp and Furuta. The construction we give here uses the action of
$\text{Mod}(S_{g,1})$ on the circle and its rotation numbers. We suspect that
our construction is the same as the construction by Furuta and Trapp using
winding numbers and provide half of the proof.
