The strong Kervaire invariant problem in dimension 62

TL;DR

This paper advances the understanding of the Kervaire invariant problem in dimension 62 by computing Toda brackets and proving the vanishing of certain elements, simplifying complex constructions in stable homotopy theory.

Contribution

It proves that heta_4^2=0 using Toda brackets, confirming the existence of heta_5 and simplifying the construction of related complexes.

Findings

heta_4^2=0 established

Existence of heta_5 confirmed

Simplified complex construction achieved

Abstract

Using a Toda bracket computation $\langle \theta_4, 2, \sigma^2\rangle$ due to Daniel C. Isaksen [11], we investigate the $45$-stem more thoroughly. We prove that $\theta_4^2=0$ using a $4$-fold Toda bracket. By [2], this implies that $\theta_5$ exists and there exists a $\theta_5$ such that $2\theta_5=0$. Based on $\theta_4^2=0$, we simplify significantly the $9$-cell complex construction in [1] to a $4$-cell complex, which leads to another proof that $\theta_5$ exists.