A four-term exact sequence of fundamental groups of orbit configuration spaces
S. K. Roushon
TL;DR
The paper studies the fundamental groups of orbit configuration spaces ${\mathcal{O}}_n(S,G)$ arising from effective, properly discontinuous actions of a discrete group on a connected aspherical 2-manifold with isolated fixed points. It relates the induced four-term exact sequence on these orbit spaces to the analogous sequence for orbifold braid groups, via orbifold covering space theory and a generalized Snake Lemma. The main result is a four-term exact sequence for $\pi_1({\mathcal{O}}_n(S,G))$ with kernel $L(S,n-1)$, and the identification $L(S,n-1)\simeq K(\overline S,n-1)$; when the action is free, $L(S,n-1)$ is trivial and the sequence simplifies. The work also shows that $K(O,n-1)$ is free for orbifolds $O$ in a specified class ${\mathcal C}$, and discusses consequences for torsion-freeness and the asphericity conjecture in non-free actions, linking orbit configuration space topology to orbifold braid group structures.
Abstract
We deduce that the fundamental groups of the orbit configuration spaces of an effective and properly discontinuous action of a discrete group on a connected aspherical 2-manifold, with isolated fixed points, fit into a four-term exact sequence. This comes as a consequence of the four-term exact sequence of orbifold braid groups ([16], [11] and [17]). The proof relates these two exact sequences and also draws a new consequence (Corollary 2.3) on the later one.