Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem

Gonçalves, Daciberg; Randall, Duane

doi:10.1016/j.crma.2006.01.016

Differential Topology

Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem
[Les auto-coincidences d'applications entre deux sphères et la forme forte du problème de Kervaire ]

Présenté par : Étienne Ghys

Gonçalves, Daciberg ¹ ; Randall, Duane 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

Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 511-513.

Résumé
Abstract

Soit $f : S^{4 n - 2} \to S^{2 n}$ une application continue entre les sphères de dimensions respectives $4 n - 2$ et $2 n$ pour $n > 4$ . Nous démontrons que, si la paire $(f, f)$ 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 $[f] \in π_{4 n - 2} (S^{2 n})$ est 1. Cette dernière condition est équivalente à une forme forte du problème de Kervaire.

Let $f : S^{4 n - 2} \to S^{2 n}$ be a map between spheres of dimensions $4 n - 2$ and $2 n$ with $n > 4$ . We show that the existence of such a map satisfying the property that the pair $(f, f) : S^{4 n - 2} \to S^{2 n}$ 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 $π_{2 n - 2}^{s}$ .

Reçu le : 2005-05-01
Accepté le : 2006-01-06
Publié le : 2006-02-23

DOI : 10.1016/j.crma.2006.01.016

Affiliations des auteurs :

Gonçalves, Daciberg ¹ ; Randall, Duane ²

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Gonçalves, Daciberg; Randall, Duane. Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem. Comptes Rendus. Mathématique, Tome 342 (2006) no. 7, pp. 511-513. doi : 10.1016/j.crma.2006.01.016. http://www.numdam.org/articles/10.1016/j.crma.2006.01.016/

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