The strong Kervaire invariant problem in dimension 62

Xu, Zhouli¹

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Room 208C, Chicago, IL 60637, USA

Geometry & topology, Tome 20 (2016) no. 3, pp. 1611-1624.

Résumé

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.

Détail

DOI : 10.2140/gt.2016.20.1611

Classification : 55Q45
Keywords: Kervaire invariant, Toda brackets

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