Hopf Invariants of Polynomial Maps and Villarceau Spheres

We prove that the stereographic images of the fibres of the complex, quaternionic and octonionic Hopf fibrations that lie on a product of two round spheres are the Villarceau spheres of a tube, which gives Villarceau’s theorem for every ring torus and its analogues in dimensions four and eight. We prove that the Hopf invariant of a holomorphic polynomial map from the three-sphere to the two-sphere is the number of common zeros of its two coordinates in the open four-ball, counted with multiplicity. For quaternionic and octonionic maps whose coordinates are one-sided polynomials in separate variables, we prove that the Hopf invariant is a sum of positive contributions of the zero points and zero spheres in the ball, read off from two real polynomials; every value from zero to the product of the degrees occurs, the formula extends to triangular maps, and positivity fails for general polynomial maps, which realise every integer. Along the way we prove that the three Hopf maps are Riemannian submersions and sphere bundles of vector bundles with Hopf invariant one, and we derive the homotopy groups of the spheres of dimensions two, four and eight from those of the spheres of dimensions three and seven.