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

Rochon, Frédéric

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

Les groupes d’homotopie du groupe (stabilisé) $G^0\left(X\right)$ 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 $G^0\left(X\right)$. 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 $G^0\left(X\right)$ 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 $G^0\left(X\right)$. 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.

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

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