On the topology of the space of invertible pseudodifferential operators of order 0

Frédéric Rochon[1]

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

Annales de l’institut Fourier (2008)

  • Volume: 58, Issue: 1, page 29-62
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Rochon, Frédéric. "Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0." Annales de l’institut Fourier 58.1 (2008): 29-62. <http://eudml.org/doc/10313>.

@article{Rochon2008,
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.},
affiliation = {State University of New York Department of Mathematics Stony Brook (USA)},
author = {Rochon, Frédéric},
journal = {Annales de l’institut Fourier},
keywords = {opérateurs pseudodifférentiels; groupes d’homotopie; $K$-théorie; déterminant résiduel; pseudodifferential operators on compact manifolds; homotopy groups; -theory},
language = {fre},
number = {1},
pages = {29-62},
publisher = {Association des Annales de l’institut Fourier},
title = {Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0},
url = {http://eudml.org/doc/10313},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Rochon, Frédéric
TI - Sur la topologie de l’espace des opérateurs pseudodifférentiels inversibles d’ordre 0
JO - Annales de l’institut Fourier
PY - 2008
PB - Association des Annales de l’institut Fourier
VL - 58
IS - 1
SP - 29
EP - 62
AB - 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.
LA - fre
KW - opérateurs pseudodifférentiels; groupes d’homotopie; $K$-théorie; déterminant résiduel; pseudodifferential operators on compact manifolds; homotopy groups; -theory
UR - http://eudml.org/doc/10313
ER -

References

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