The strong Kervaire invariant problem in dimension $62$

Abstract

Using a Toda bracket computation $\langle {\theta }_4,2,{\sigma }^2\rangle$ due to Daniel C Isaksen, we investigate the $45$–stem more thoroughly. We prove that ${\theta }_4^2=0$ using a $4$–fold Toda bracket. By work of Barratt, Jones and Mahowald, 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 their $9$–cell complex construction to a $4$–cell complex, which leads to another proof that ${\theta }_5$ exists.