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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?

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    $\begingroup$ Already the case of two points seems interesting. If $X$ is a smooth manifold then looking at the fibers of $FM(X,2) \to X^2$ over the diagonal gives you back the projectivized tangent bundle of $X$. If the construction of FM worked also when $X$ is not smooth, what would you get then? Can one make sense of a "projectivized microbundle"? $\endgroup$ Oct 20, 2020 at 16:12
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    $\begingroup$ @DanPetersen PL (and top) manifolds have regular neighborhoods, which are like the total space of disk bundles, but the map to the base space isn't a fiber bundle. For $n=2$ you can just take the complement of the interior of the regular neighborhood. This is probably as good as it gets. It is defined up to a contractible choice. Another contractible choice can arrange for it to surject to $X^2$ with fibers the right homotopy type. That is probably possible for general $n$. $\endgroup$ Oct 20, 2020 at 16:36
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    $\begingroup$ Inspired by what Dan Petersen said, if $FM(X,2)$ can be constructed functorially, there would be a group homomorphism $\mathrm{Homeo}^+(\mathbb{R}^2,0) \to \mathrm{Homeo}^+(S^1)$ inducing the usual weak equivalence on classifying spaces. Maybe that can be ruled out somehow? Katie Mann and others have studied homomorphisms between homeomorphisms groups. $\endgroup$
    – user169545
    Nov 28, 2020 at 16:40

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In this note I used recent results of Chen and Mann rule out the existence of such a topological compactification in all dimensions $\geq 2$.

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