Homotopy properties of regular mappings into real retract rational varieties

TL;DR

This paper investigates the homotopy classes of regular maps from spheres into real retract rational varieties, demonstrating subgroup structures and constructing representatives for Whitehead products, thus advancing understanding of their topological properties.

Contribution

It establishes that homotopy classes of regular maps form subgroups of homotopy groups and provides explicit regular representatives for all Whitehead products in these groups.

Findings

Homotopy classes form subgroups of homotopy groups.

Homotopy groups are independent of basepoint in connected varieties.

Constructs regular representatives for all Whitehead products.

Abstract

We study homotopy properties of regular mappings from spheres into a real retract rational variety $Y$. We show that the homotopy classes which are represented by such mappings form subgroups of the homotopy groups of $Y$, and that the groups are independent of the choice of the basepoint on $Y$ as long as $Y$ is connected. We also construct regular representatives of all the Whitehead products in all the homotopy groups of $Y$.