On rectangular Toda brackets and Oda's extension problems

TL;DR

This paper introduces the rectangular Toda bracket as a new tool to solve longstanding extension problems in unstable homotopy groups of spheres, specifically addressing cases that have remained unresolved for over forty years.

Contribution

The paper presents the rectangular Toda bracket, a novel method that successfully resolves N. Oda's extension problems for certain homotopy groups of spheres.

Findings

Resolved extension problems for rac{39}{ ext{th}}$, rac{40}{ ext{th}}$, and rac{41}{ ext{st}}; homotopy groups of spheres.

Introduced the rectangular Toda bracket as a new computational tool.

Demonstrated the effectiveness of the method in longstanding unresolved cases.

Abstract

This paper tackles \textit{N. Oda}'s extension problems for the homotopy groups $\pi_{39}(S^{6})$, $\pi_{40}(S^{7})$, and $\pi_{41}(S^{8})$ localized at 2, the issues having eluded resolution for more than four decades. We introduce a tool for the theory of determinations of unstable homotopy groups, namely, the rectangular Toda bracket, by which we are able to solve the extension problems with respect to these three homotopy groups.