  In the framework of fibred cusp operators on a manifold $X$ associated to a
boundary fibration $\Phi: \pa X\to Y$, the homotopy groups of the space 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 the K-theory of $T^{*}Y$.
