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

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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
##### Publication
Received: 4 November 2014
Accepted: 18 July 2015
Published: 4 July 2016
Proposed: Paul Goerss
Seconded: Haynes Miller, Ronald Stern
##### Authors
 Zhouli Xu Department of Mathematics University of Chicago 5734 S. University Avenue, Room 208C Chicago, IL 60637 USA