Homotopy types of spaces of submanifolds of Rⁿ

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We compute the homotopy type of the space of proper d-dimensional submanifolds of Rⁿ with a smooth version of the Fell topology. Our methods allow us to compute the homotopy type of the space of submanifolds with summable labels too, and to give a new proof of the Galatius--Randal-Williams theorem on the homotopy type of their space of submanifolds