The strong Kervaire invariant problem in dimension 62

Xu, Zhouli

doi:10.2140/gt.2016.20.1611

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The strong Kervaire invariant problem in dimension $62$

Zhouli Xu

Geometry & Topology 20 (2016) 1611–1624

DOI: 10.2140/gt.2016.20.1611

Abstract

Using a Toda bracket computation $〈 θ_{4}, 2, σ^{2} 〉$ due to Daniel C Isaksen, we investigate the $45$ –stem more thoroughly. We prove that $θ_{4}^{2} = 0$ using a $4$ –fold Toda bracket. By work of Barratt, Jones and Mahowald, this implies that $θ_{5}$ exists and there exists a $θ_{5}$ such that $2 θ_{5} = 0$ . Based on $θ_{4}^{2} = 0$ , we simplify significantly their $9$ –cell complex construction to a $4$ –cell complex, which leads to another proof that $θ_{5}$ exists.

Keywords

Kervaire invariant, Toda brackets

Mathematical Subject Classification 2010

Primary: 55Q45

References

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

Volume 20, issue 3 (2016)