# $$S^1$$-Equivariant Dirac operators on the Hopf Fibration
In this post I want to present some notes around the fundamentals of the Hopf fibration $$\pi: S^3 \longrightarrow S^2$$ which I started writing during my PhD period at Humboldt Universität zu Berlin. First, I describe its definition and I show that it is a non-trivial map by computing its first Chern class. Then I give a detailed treatment of the construction of two important Dirac-type operators: the Hodge-de Rham and the spin-Dirac. As these two operators commute with the $$S^1$$-action, I describe how to push them down to $$S^2$$ through the orbit map $$\pi$$, inspired by the work of Brüning and Heintze.