Homotopy groups of diagonal complements

Abstract

For $X$ a connected finite simplicial complex we consider ${\Delta }^d\left(X,n\right)$, the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d\left(X,n\right)$, the analogous space of configurations of unordered points. These reduce to the standard configuration spaces of distinct points when $d=1$. We describe the homotopy groups of ${\Delta }^d\left(X,n\right)$ (resp. $B^d\left(X,n\right)$) in terms of the homotopy (resp. homology) groups of $X$ through a range which is generally sharp. It is noteworthy that the fundamental group of the configuration space $B^d\left(X,n\right)$ abelianizes as soon as we allow points to collide, ie $d\ge 2$.