Roots of maps between spheres and projective spaces in codimension one

TL;DR

This paper classifies homotopy classes of maps from spheres and projective spaces into spheres and projective spaces, and determines the minimal root sets for non null-homotopic maps, showing they are either a circle or two circles.

Contribution

It provides a complete classification of homotopy classes and characterizes the minimal root sets for these maps in codimension one.

Findings

Minimal root set for non null-homotopic maps is either a circle or two circles.

Classification of homotopy classes of maps from $S^3$ and $ ext{RP}^3$ into $S^2$ and $ ext{RP}^2$.

Minimal root sets depend on the target space, being a single circle or two circles.

Abstract

For maps from $S^3$ and $\RP^3$ into $S^2$ and $\RP^2$, we study the problem of minimizing the root set by deforming the maps through homotopies. After presenting the classification of the homotopy classes of such maps, we prove that the minimal root set for a non null-homotopic map is either a circle or the disjoint union of two circle, according its range is $S^2$ or $\RP^2$, respectively.