Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0  [ On the topology of the space of invertible pseudodifferential operators of order 0 ]
Annales de l'Institut Fourier, Volume 58 (2008) no. 1, p. 29-62

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.

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.

DOI : https://doi.org/10.5802/aif.2343
Classification:  58B05,  58B15
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     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, Volume 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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