Homotopy properties of regular mappings into real retract rational varieties

TL;DR

The paper studies algebraic counterparts of homotopy classes of maps from spheres into real nonsingular retract rational varieties. It develops a rational H-space framework and combines it with approximation theorems to show that algebraic homotopy classes form subgroups of classical homotopy groups, are basepoint-independent on connected $Y$, and have regular representatives for Whitehead products; it also proves an approximation result for maps from nonsingular affine curves and presents counterexamples illustrating the limits of generalization. Together, these results advance the understanding of how real algebraic geometry constrains homotopy-theoretic phenomena and indicate the finely balanced nature of retract rational varieties in this context.

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 $Y$.

Theorems & Definitions (23)

• Definition 1.1
• Theorem 1.2: baneckiRetractRationalVarieties2025
• Theorem 1.3
• Theorem 1.4
• Remark 1.4.1
• proof
• Theorem 1.6
• Theorem 1.7
• Corollary 1.8
• proof