Non-factorisation of Arf-Kervaire classes through ${\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$

TL;DR

This paper demonstrates that certain Arf-Kervaire classes cannot be factored through the smash product of infinite real projective spaces, using advanced homotopy theoretical methods.

Contribution

It applies the upper triangular technology method to prove non-existence of specific stable homotopy classes related to Arf-Kervaire invariant one.

Findings

No stable homotopy classes of ${ m RP}^ fty imes { m RP}^ fty$ exist in certain dimensions with Arf-Kervaire invariant one.

The result restricts possible factorizations of Arf-Kervaire classes.

The proof employs the upper triangular technology method of Snaith.

Abstract

As an application of the upper triangular technology method of (V.P. Snaith: {\em Stable homotopy -- around the Arf-Kervaire invariant}; Birkh\"{a}user Progress on Math. Series vol. 273 (April 2009)) it is shown that there do not exist stable homotopy classes of $ {\mathbb RP}^{\infty} \wedge {\mathbb RP}^{\infty}$ in dimension $2^{s+1}-2$ with $s \geq 2$ whose composition with the Hopf map to $ {\mathbb RP}^{\infty}$ followed by the Kahn-Priddy map gives an element in the stable homotopy of spheres of Arf-Kervaire invariant one.