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Equivalences D'Homotopie et Crochet de Whitehead

Published online by Cambridge University Press:  20 November 2018

G. Didierjean  and
A. Legrand
G. Didierjean
Affiliation:
IRMA Strasbourg URA CNRS 1 Université Louis Pasteur 7 rue René Descartes F-67084 Strasbourg, cédex France
A. Legrand
Affiliation:
Laboratoire de Topologie et Géométrie URA CNRS 1408 Université Paul Sabatier 118 route de Narbonne F-31062 Toulouse, cédex France
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Résumé

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L'homotopie de l'espace des équivalences homotopie fibrées, est limite d'une suite spectrale dont on calcule ici la première différentielle. On montre, sous des hypothèses assez générales et dans le cas où le fibré admet une section, que cette différentielle est la somme des trois opérations suivantes:

une opération produit tensoriel d'une opération cohomologique type carré de Steenrod avec une opération homotopique de Hopf

une opération définie par le "Brace-product" du fibré

une opération définie par les crochets de Whitehead de la fibre

La différentielle de la suite spectrale associée à l'homotopie de l'espace des équivalences d'homotopie d'un espace s'obtient en prenant la base réduite à un point. Ce calcul prolonge certains résultats de [KA 69], [CO-HA].

Abstract

Abstract

We compute the first differential of a spectral sequence which converges to the homotopy of the fiberwise self homotopy equivalences space of a bundle. With sufficiently general hypotheses and when the bundle is equiped with a cross section, we show that this differential is the sum of three operations

an operation tensor product of a cohomology operation like Steenrod square with an Hopf homotopy operation

an operation associated to the Brace product of the bundle

an operation associated to the Whitehead bracket of the fibre

Taking a one point set base space we obtain the differential of the spectral sequence related to the homotopy of the self homotopy equivalences space of a space. This extends some results of [KA 69], [CO-HA].

Keywords

55P1055T99homotopy equivalencespectral sequencePostnikov tower
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 46 , Issue 2 , 01 April 1994 , pp. 253 - 273
Copyright
Copyright © Canadian Mathematical Society 1994

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