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](https://arxiv.org/pdf/math/0306385.pdf) by Dev Sinha for details and an overview of the rich history of this construction.



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> 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?
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> 
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A piecewise-linear or topological Fulton-MacPherson compactification
In [this note](https://arxiv.org/abs/2011.14855) I used recent results of Chen and Mann rule out the existence of such a topological compactification in all dimensions $\geq 2$.


