Computing the Topological Degree of Maps Between 2-Spheres
Daniil Kucher
TL;DR
This work addresses the problem of computing the topological degree of maps $R:S^2\to S^2$ by generalizing the rational-degree framework to arbitrary continuous $f,g$ without common zeros. It represents $R$ as $f/g$ and leverages the Hopf fibration to reduce degree computation to a winding number on $\mathbb{C}\setminus\{0\}$, enabling effective evaluation via a loop $\tilde{f}(\phi)=f(Me^{i\phi})$. In the polynomial-in-$z,\bar{z}$ setting, the degree is given by counting roots of the homogeneous top-degree term via $\tilde{T}(z)=z^d T(z,z^{-1})$, with the final count adjusted by $-d$ when appropriate. The method remains valid under a finite/infinite limit at infinity (via Möbius conjugation that preserves degree) and is demonstrated through explicit computational examples, providing practical algorithms for degrees of maps between 2-spheres.
Abstract
We describe an effective method for computing the topological degree of continuous functions $R:S^2 \to S^2$, where $S^2$ is the Riemann sphere. Our approach generalizes the degree formula for rational functions of complex polynomials, $\frac{f}{g}$, without common zeros. To apply our method, it is necessary to represent the function $R$ as the ratio of two continuous complex-valued functions $f$ and $g$ without common zeros. By using the Hopf fibration, this method reduces the problem to computing the winding number of a loop. This enables us to compute the degree of $\frac{f}{g}$ even when $f$ and $g$ are arbitrary continuous complex functions without common zeros, and the fraction has a limit at infinity (which can be finite or infinite). Specifically, if $f$ and $g$ are complex polynomials in $z$ and $\bar{z}$, and the highest-degree homogeneous component of the polynomial with the greater algebraic degree has a finite or infinite limit as $|z|\to\infty$, then the problem reduces to counting the roots of a complex polynomial inside the unit circle, obtained from this component.