Quasifibrations in configuration Lie groupoids

Abstract

In [18] we studied a Fadell-Neuwirth type fibration theorem for orbifolds, and gave a short exact sequence of fundamental groups of configuration Lie groupoids of Lie groupoids corresponding to the genus zero 2-dimensional orbifolds with cone points, and at least one puncture. In this paper we extend this work to all genus > 0, 2-dimensional orbifolds with cone points. As a consequence, we prove the Farrell-Jones Isomorphism conjecture for the fundamental groups of the associated configuration Lie groupoids. This answers a substantial part of a question we posed in [[17], Problem]. In [18] we also showed that for all global quotient type orbifolds, the fibration theorem does not hold. Here, we provide some nontrivial examples of orbifolds where a Fadell-Neuwirth type quasifibration theorem holds. Finally, we state an Asphericity conjecture and a Quasifibration conjecture for orbifolds.