Bott periodicity for fibred cusp operators

In the framework of fibred cusp operators on a manifold X associated to a boundary fibration Φ: ∂X → Y, the homotopy groups of the space Gφ−∞ (X; E) of invertible smoothing perturbations of the identity are computed in terms of the K-theory of T*Y. It is shown that there is a periodicity, namely the odd and the even homotopy groups are isomorphic among themselves. To obtain this result, one of the important steps is the description of the index of a Fredholm smoothing perturbation of the identity in terms of an associated K-class in Kc0(T*Y).