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

TL;DR

This paper establishes a bound on the complexity of essential simplicial maps from 3-spheres to 2-spheres, demonstrating the minimality of a specific Hopf map triangulation within its homotopy class.

Contribution

It proves a fiber-uniform bound on the complexity of essential simplicial maps from S^3 to S^2 and shows the minimality of a particular Hopf map triangulation.

Findings

Bound on the complexity of essential simplicial maps from S^3 to S^2

The constructed Hopf map triangulation is minimal in its homotopy class

Investigation of the tightness of the complexity bound

Abstract

A fiber-uniform bound on the complexity of an essential simplicial map $S^3\rightarrow S^2$ 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 $S^3$.