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 constructions 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 also show that our construction is the same as the construction by Furuta and Trapp using winding numbers.
Categories: math.GT