## Geometry & Topology

### The strong Kervaire invariant problem in dimension $62$

Zhouli Xu

#### Abstract

Using a Toda bracket computation $〈θ4,2,σ2〉$ due to Daniel C Isaksen, we investigate the $45$–stem more thoroughly. We prove that $θ42 = 0$ using a $4$–fold Toda bracket. By work of Barratt, Jones and Mahowald, this implies that $θ5$ exists and there exists a $θ5$ such that $2θ5 = 0$. Based on $θ42 = 0$, we simplify significantly their $9$–cell complex construction to a $4$–cell complex, which leads to another proof that $θ5$ exists.

#### Article information

Source
Geom. Topol., Volume 20, Number 3 (2016), 1611-1624.

Dates
Accepted: 18 July 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858998

Digital Object Identifier
doi:10.2140/gt.2016.20.1611

Mathematical Reviews number (MathSciNet)
MR3523064

Zentralblatt MATH identifier
1352.55007

Subjects
Primary: 55Q45: Stable homotopy of spheres

Keywords
Kervaire invariant Toda brackets

#### Citation

Xu, Zhouli. The strong Kervaire invariant problem in dimension $62$. Geom. Topol. 20 (2016), no. 3, 1611--1624. doi:10.2140/gt.2016.20.1611. https://projecteuclid.org/euclid.gt/1510858998

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