Geometric approach to stable homotopy groups of spheres, III. Abelian, cyclic and quaternionic structure for mappings with singularities

TL;DR

This paper develops a geometric framework for understanding the stable homotopy groups of spheres by constructing PL-mappings with specific algebraic structures such as abelian, cyclic, and quaternionic, to analyze their properties.

Contribution

It introduces a geometric approach to classify stable homotopy groups using PL-mappings with various algebraic structures, expanding the understanding of their topological and algebraic properties.

Findings

Constructed PL-mappings with abelian, cyclic, and quaternionic structures.

Established relationships between these structures and stable homotopy groups.

Provided new tools for analyzing mappings with singularities in topology.

Abstract

Collection of (equivariant) $\rm{PL}$-mappings admitting a relative abelian, cyclic, quaternionic, bicyclic, and quaternionic-cyclic structures are constructed.