Circle action of the punctured mapping class group and cross homomorphism

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.