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In the following short note, we give a new geometric interpretation of thegenerator of the infinite cyclic group$H^1(\text{Mod}(S_{g,1});H^1(S_g;\mathbb{Z}))$ (this computation is proved byMorita). 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 thatour construction is the same as the construction by Furuta and Trapp usingwinding numbers and provide half of the proof.

Circle action of the punctured mapping class group and cross homomorphism