  We study the analytic and homotopy properties of some infinite dimensional
Grassmannians, useful for developing a Morse theory for infinite dimensional
manifolds. We study the space of Fredholm pairs of a Hilbert space, we
determine its homotopy type, and we define a determinant bundle over it. We
study the space of compact perturbations of a given closed linear subspace, and
the related concept of essential Grassmannian.
