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

Abstract

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

5 pages

References in corpus (3)

• Geometric approach towards stable homotopy groups of spheres. The Kervaire invariant II
• The Arf-Kervaire Invariant of framed manifolds
• Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I