The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. You can construct them by iterated spherical blow-ups, or directly as the closure of the image of a certain map. See this paper by Dev Sinha for details and an overview of the rich history of this construction.

Is it possible to construct the Fulton-MacPherson compactification up to (PL-)homeomorphism without reference to a smooth structure. That is, can we give a construction that also works for piecewise-linear or topological manifolds?