Homotopy Groups of Diagonal Complements

TL;DR

This paper investigates the homotopy groups of generalized configuration spaces where no more than d points coincide, revealing that allowing collisions simplifies the fundamental group to an abelian group for d≥2.

Contribution

It provides explicit descriptions of the homotopy groups of diagonal complement configuration spaces in terms of the base space's homotopy and homology groups, with sharp ranges.

Findings

Homotopy groups of diagonal complement configuration spaces are characterized.

Fundamental groups become abelian when points are allowed to collide (d≥2).

Descriptions are sharp within a specific range.

Abstract

For $X$ a connected finite simplicial complex we consider $\Delta^d(X,n)$ the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d(X,n)$ 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 $\Delta^d(X,n)$ (resp. $B^d(X,n)$) 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 $B^d(X,n)$ abelianizes as soon as we allow points to collide (i.e. $d\geq 2$).