Let X be a CW-complex; we shall consider the group 2 s[x]formed by the homotopy classes of equivalence maps from X into itself with the operation induced by the composition of maps. It is clear to see that this group depends only on the homotopy type of X, hence should be determined by the known homotopy invariants of X. This is the problem which we shall try to study here. In fact, there exists a spectral sequence converging to S[X], whose initial terms are given, roughly speaking, by the cohomology of X and the automorphism group of its homotopy group.Besides the satisfaction of curiosity, the group S[X] seems to have other interests. For example, it operates canonically on the special cohomology group [l] of X

There is a relation between Hirzebruch's differentiable RiemannRoch theorem [12]

In [l] Hörmander defines the generalized symbol of a pseudodifferential operator P as a sequence of partially defined maps between function spaces. Our purpose here is to comment on the existence of characteristic polynomial type symbols

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Sur les complexes généralisés et la suite spectrale d'un fibré Bulletin de la S. M. F., tome 87 (1959), p. 451-453 © Bulletin de la S. M. F., 1959, tous droits réservés. L'accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/