A note on the divisibility of the Whitehead square

TL;DR

This paper investigates the divisibility properties of Whitehead squares in stable homotopy groups, linking torsion assumptions to the existence of Kervaire invariant one elements in specific dimensions.

Contribution

It establishes a divisibility condition for Whitehead squares under torsion-free assumptions and applies known results to confirm the existence of Kervaire invariant one elements in dimensions 62 and 126.

Findings

Whitehead squares are divisible by 2 under certain torsion-free conditions.

Existence of Kervaire invariant one elements in dimensions 62 and 126 confirmed.

Provides a link between torsion properties and Kervaire invariant elements.

Abstract

We show that if we suppose n>3 and the (2n-1)-stem in the stable homotopy groups of spheres has no 2-torsion, then the Whitehead squares of the identity maps of (2n+1) and (4n+3)-spheres are divisible by 2. Applying the result of G. Wang and Z. Xu on the 61-stem in the stable homotopy groups of spheres, we find that the Kervaire invariant one elements in dimensions 62 and 126 exist.