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

Daciberg Gonçalves; Duane Randall

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

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

Comptes Rendus. Mathématique, Volume 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 :

Daciberg Gonçalves ¹ ; Duane Randall ²

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     title = {Self-coincidence of mappings between spheres and the {Strong} {Kervaire} {Invariant} {One} {Problem}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {511--513},
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     language = {en},
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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/

Bibliographie
Cité par

[1] M. Barratt; J. Jones; M. Mahowald The Kervaire invariant problem, Contemp. Math., Volume 19 (1983), pp. 9-22

[2] M. Barratt; J. Jones; M. Mahowald The Kervaire invariant and the Hopf invariant, Lecture Notes in Math., vol. 1286, Springer-Verlag, 1987, pp. 135-173

[3] W.E. Browder The Kervaire invariant of framed manifolds and its generalizations, Ann. of Math., Volume 90 (1969), pp. 157-186

[4] F.R. Cohen Fibration and product decompositions in nonstable homotopy theory, Handbook of Algebraic Topology, North-Holland, 1995, pp. 1175-1208

[5] A. Dold; D.L. Gonçalves Self-coincidence of fibre maps, Osaka J. Math., Volume 42 (2005), pp. 291-307

[6] D.L. Gonçalves Coincidence theory (R.F. Brown; M. Furi; L. Górniewicz; B. Jiang, eds.), Handbook of Topological Fixed Point Theory, Springer, 2005, pp. 1-42

[7] D. Gonçalves, D. Randall, Self-coincidence of maps from $S^{q}$ -bundles over $S^{n}$ to $S^{n}$ , Bol. Soc. Mexicana Mat. 10 (3) (2004) 181–192 (special issue)

[8] D. Gonçalves, D. Randall, Self-coincidence of maps between spheres, in preparation

[9] U. Koschorke Selfcoincidences in higher codimensions, J. Reine Angew. Math., Volume 576 (2004), pp. 1-10

[10] V. Snaith; J. Tornehave On $π_{*}^{s} (B O)$ and the Arf invariant of framed manifolds, Contemp. Math., Volume 12 (1982), pp. 299-314

[11] H. Toda Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Stud., vol. 49, Princeton Univ. Press, Princeton, 1962

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