Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0

Rochon, Frédéric

doi:10.5802/aif.2343

Rochon, Frédéric ¹

¹ State University of New York Department of Mathematics Stony Brook (USA)

Annales de l'Institut Fourier, Tome 58 (2008) no. 1, pp. 29-62.

Résumé
Abstract

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.

MR Zbl

DOI : 10.5802/aif.2343

Classification : 58B05, 58B15
Mots-clés : opérateurs pseudodifférentiels, groupes d’homotopie, $K$-théorie, déterminant résiduel

Affiliations des auteurs :

Rochon, Frédéric ¹

¹ State University of New York Department of Mathematics Stony Brook (USA)

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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/articles/10.5802/aif.2343/

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