Differential Topology
Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem
Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 511-513.

Let $f\phantom{\phantom{\rule{0.2em}{0ex}}}:{S}^{4n-2}\to {S}^{2n}$ be a map between spheres of dimensions $4n-2$ and $2n$ with $n>4$. We show that the existence of such a map satisfying the property that the pair $\left(f,f\right)\phantom{\phantom{\rule{0.2em}{0ex}}}:{S}^{4n-2}\to {S}^{2n}$ can be deformed to a coincidence free pair but cannot be deformed to coincidence free by small deformation is equivalent to the Strong Kervaire Invariant One Problem, i.e., the existence of an element of order 2 with Kervaire invariant one in the stable homotopy group ${\pi }_{2n-2}^{s}$.

Soit $f\phantom{\phantom{\rule{0.2em}{0ex}}}:{S}^{4n-2}\to {S}^{2n}$ une application continue entre les sphères de dimensions respectives $4n-2$ et $2n$ pour $n>4$. Nous démontrons que, si la paire $\left(f,f\right)$ est déformable en une paire libre de coïncidences, alors elle n'est pas déformable par petites déformations si et seulement si $n={2}^{j}$, $j⩾3$, et l'invariant de Kervaire de la classe d'homotopie $\left[f\right]\in {\pi }_{4n-2}\left({S}^{2n}\right)$ est 1. Cette dernière condition est équivalente à une forme forte du problème de Kervaire.

Accepted:
Published online:
DOI: 10.1016/j.crma.2006.01.016

Daciberg Gonçalves 1; Duane Randall 2

1 Departamento de Matemática, IME, USP, Caixa Postal 66.281, CEP, 05311-970, São Paulo, SP, Brazil
2 Department of Mathematics and Computer Science, Loyola University, 6363 St. Charles Avenue, New Orleans, LA 70118, USA
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Daciberg Gonçalves; Duane Randall. Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem. Comptes Rendus. Mathématique, Volume 342 (2006) no. 7, pp. 511-513. doi : 10.1016/j.crma.2006.01.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.016/

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