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.

The following is an interesting animation of the Hopf fibration I found online:

— Vignette (@vi_ne_te) September 5, 2020