We study the differentiable structure and the homotopy type of some spaces related to the Grassmannian of closed linear subspaces of an infinite dimensional Hilbert space, such as the space of Fredholm pairs, the Grassmannian of compact perturbations of a given space, and the essential Grassmannians. We define a determinant bundle over the space of Fredholm pairs. We lift the composition of Fredholm operators to the Quillen determinant bundle, and we show how this map can be used in several constructions involving the determinant bundle over the space of Fredholm pairs. We deduce some properties of suitable orientation bundles.
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