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Take a path-connected space $X$ and consider the family of non-empty, ordered configuration spaces $(\text{Conf}_{\bullet+1}(X))$. This collection of spaces forms a semi-simplicial space with face maps $d_i =$ forget the $i$-th point. This can be seen as a subspace of the family of cartesian powers $(X^{\times (\bullet+1)})$ with the same face maps (here, there are also degeneracies).

It is a know fact that the realization of the latter simplicial space is contractible : it is the same proof that the simplicial construction of $EG$ is contractible for a group $G$.

One could construct degeneracy maps for the configuration simplicial space by considering Fulton-MacPherson compactification of these configuration spaces (when $X$ is a manifold), and the existence of a non-vanishing vector field on $X$ (to choose how to put an additional point "infinitesimally close" to another).

My question is : do we know the homotopy type of the (fat) realization $|\text{Conf}_{\bullet+1}(X)|$ ? Is it weakly contractible as it is the case for $|X^{\times (\bullet+1)}|$?

Nicolas Guès
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Wahl considers a framed analog of this question in the case of surfaces in "Homological Stability for mapping class groups of surfaces". On page 28, she shows this semisimplicial object is contractible when the surface has boundary components. The dimension and the framing do not appear to play a role in the argument, though I have not carefully checked. From this fact she then deduces the case of a closed surface which also does not seem to use these facts.

More general statements can be found in section 11.2 of Randall-William's "Resolutions of moduli spaces and homological stability" and I believe in the case the manifold is framed, you can play around with tangential structures to directly deduce the result you are after from the version with framings.

Connor Malin
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