The Kustaanheimo–Stiefel map, the Hopf fibration and the square root map on R3 and R4

We study the Kustaanheimo–Stiefel map (KSM) ψ from U∗:=R4∖{0} to X∗:=R3∖{0} and the principal circle bundle P=(U∗,ψ,X∗,S1) that it induces. We show that the KSM is the appropriate generalization of the squaring map z↦z2, z∈C, and not quaternion-multiplication, in that the KSM induces a principal circle bundle on S3→S2, namely the Hopf fibration, while quaternion-squaring is degenerate because the dimension of the fibers is not constant.We construct two square root branches from the upper and lower half of R3 to R3∖−(x1) where −(x1) is the nonpositive x1-axis in R3 and resembles the cut used to define the standard complex square root branches . We glue these two branches together.We introduce what we like to call KS cylindrical coordinates with a 2-dimensional axis of rotation. We also introduce what we call KS torical and spherical coordinates.We use the KS cylindrical coordinates to define the full square root map on an S1-cover of R3 given by (R3×S1)/∼, where ∼ is an equivalence relation on −(x1)×S1.