Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0
Annales de l'Institut Fourier, Tome 58 (2008) no. 1, p. 29-62
Les groupes d’homotopie du groupe (stabilisé) G 0 (X) des opérateurs pseudodifférentiels inversibles d’ordre zéro agissant sur une variété compacte sans bord X sont calculés en termes de la K-théorie du fibré cosphérique S * X. Du même coup, on montre que le sous-groupe des perturbations compactes inversibles de l’identité est faiblement rétractile dans G 0 (X). Les résultats sont aussi adaptés au cas des opérateurs suspendus. Des applications à la théorie de l’indice et pour le déterminant résiduel de Simon Scott sont aussi données.
The homotopy groups of the (stabilized) group G 0 (X) of invertible pseudodifferential operators of order zero acting on a smooth compact manifold X are given in terms of the K-theory of the cosphere bundle S * X. At the same time, it is shown that the subgroup of invertible compact perturbations of the identity is weakly retractible in G 0 (X). The results are also adapted to the case of suspended operators. This gives applications in index theory and for the residue determinant of Simon Scott.
DOI : https://doi.org/10.5802/aif.2343
Classification:  58B05,  58B15
Mots clés: opérateurs pseudodifférentiels, groupes d’homotopie, K-théorie, déterminant résiduel
@article{AIF_2008__58_1_29_0,
     author = {Rochon, Fr\'ed\'eric},
     title = {Sur la topologie de l'espace des op\'erateurs pseudodiff\'erentiels inversibles d'ordre 0},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {1},
     year = {2008},
     pages = {29-62},
     doi = {10.5802/aif.2343},
     mrnumber = {2401215},
     zbl = {1154.58014},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2008__58_1_29_0}
}
Rochon, Frédéric. Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0. Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 29-62. doi : 10.5802/aif.2343. http://www.numdam.org/item/AIF_2008__58_1_29_0/

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