On essential simplicial maps $S^3\to S^2$

A fiber-uniform bound on the complexity of an essential simplicial map $S^3\to S^2$ is proven, and the tightness of the bound is investigated. It follows that the triangulation of the Hopf map constructed by Madahar and Sarkaria is minimal in its homotopy class in terms of the number of 3-simplices in the triangulation of $S^3$.