Let be a map between spheres of dimensions and with . We show that the existence of such a map satisfying the property that the pair 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 .
Soit une application continue entre les sphères de dimensions respectives et pour . Nous démontrons que, si la paire est déformable en une paire libre de coïncidences, alors elle n'est pas déformable par petites déformations si et seulement si , , et l'invariant de Kervaire de la classe d'homotopie est 1. Cette dernière condition est équivalente à une forme forte du problème de Kervaire.
Accepted:
Published online:
Daciberg Gonçalves 1; Duane Randall 2
@article{CRMATH_2006__342_7_511_0, author = {Daciberg Gon\c{c}alves and Duane Randall}, title = {Self-coincidence of mappings between spheres and the {Strong} {Kervaire} {Invariant} {One} {Problem}}, journal = {Comptes Rendus. Math\'ematique}, pages = {511--513}, publisher = {Elsevier}, volume = {342}, number = {7}, year = {2006}, doi = {10.1016/j.crma.2006.01.016}, language = {en}, }
TY - JOUR AU - Daciberg Gonçalves AU - Duane Randall TI - Self-coincidence of mappings between spheres and the Strong Kervaire Invariant One Problem JO - Comptes Rendus. Mathématique PY - 2006 SP - 511 EP - 513 VL - 342 IS - 7 PB - Elsevier DO - 10.1016/j.crma.2006.01.016 LA - en ID - CRMATH_2006__342_7_511_0 ER -
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/
[1] The Kervaire invariant problem, Contemp. Math., Volume 19 (1983), pp. 9-22
[2] The Kervaire invariant and the Hopf invariant, Lecture Notes in Math., vol. 1286, Springer-Verlag, 1987, pp. 135-173
[3] The Kervaire invariant of framed manifolds and its generalizations, Ann. of Math., Volume 90 (1969), pp. 157-186
[4] Fibration and product decompositions in nonstable homotopy theory, Handbook of Algebraic Topology, North-Holland, 1995, pp. 1175-1208
[5] Self-coincidence of fibre maps, Osaka J. Math., Volume 42 (2005), pp. 291-307
[6] 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 -bundles over to , 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] Selfcoincidences in higher codimensions, J. Reine Angew. Math., Volume 576 (2004), pp. 1-10
[10] On and the Arf invariant of framed manifolds, Contemp. Math., Volume 12 (1982), pp. 299-314
[11] Composition Methods in Homotopy Groups of Spheres, Ann. of Math. Stud., vol. 49, Princeton Univ. Press, Princeton, 1962
Cited by Sources:
Comments - Policy