Sur la topologie de l'espace des operateurs pseudodifferentiels inversibles d'ordre 0

The homotopy groups of the (stabilized) group of invertible pseudodifferential operators of order zero acting on a closed manifold X are computed in terms of the K-theory of the cosphere bundle S*X. At the same time, we show that the subgroup of invertible compact perturbations of the identity is weakly retractable inside this group. These results are also adapted to the case of suspended operators. Some applications in index theory and for the residue determinant of Simon Scott are also given.