#### Volume 20, issue 3 (2016)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
The strong Kervaire invariant problem in dimension $62$

### Zhouli Xu

Geometry & Topology 20 (2016) 1611–1624
##### Abstract

Using a Toda bracket computation $〈{\theta }_{4},2,{\sigma }^{2}〉$ 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.

##### Keywords
Kervaire invariant, Toda brackets
Primary: 55Q45