Homotopy types of spaces of submanifolds of ${\mathbb{R}}^n$

We compute the homotopy type of the space of proper d-dimensional submanifolds of ${\mathbb{R}}^n$ 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