 Surjective degree-zero sphere-to-torus map

On the unit two-sphere this formula maps onto the torus: every torus point has arguments corresponding to $x,y\in [-1/2,1/2]$, and $x^2+y^2\le 1/2$ leaves a real $z$ on the sphere. Scaling both arguments continuously to zero is a null-homotopy, so the degree is zero. Surjectivity alone does not force nonzero degree.