For a connected finite simplicial complex we consider , the space of configurations of ordered points of such that no of them are equal, and , the analogous space of configurations of unordered points. These reduce to the standard configuration spaces of distinct points when . We describe the homotopy groups of (resp. ) in terms of the homotopy (resp. homology) groups of through a range which is generally sharp. It is noteworthy that the fundamental group of the configuration space abelianizes as soon as we allow points to collide, ie .
"Homotopy groups of diagonal complements." Algebr. Geom. Topol. 16 (5) 2949 - 2980, 2016. https://doi.org/10.2140/agt.2016.16.2949