Homotopy Groups of Diagonal Complements

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. $d\ge 2$).

Major improvement from v1. A more careful use of Smale's result 6.1 has led to a new section 5. The main improvement is in the bound for Theorem 1.2 which is now sharp. This paper is to appear in AGT. 20 Pages

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