Manifold covers of 3-orbifolds with geometric pieces
The notion of orbifold, or V-manifold, was introduced by Satake and Thurston to describe the local structure of properly discontinuous actions on manifolds. An orbifold consists of a topological space together with an atlas which locally identifies this topological space to the quotient space of a finite group action on a manifold (plus a little more local information). Typical examples of orbifolds arise from quotient spaces of properly discontinuous actions on manifolds; such orbifolds are said to be good. Exactly which 2-dimensional orbifolds are good is very well understood. In dimension 3, there are two basic conjectures. The first one asks whether a 3-dimensional orbifold is good if and only if all of its 2-suborbifolds are good. The second one asserts that every compact good 3-orbifold can be built from pieces admitting geometric structures, as in Thurston's program to understand 3-manifolds [\textit{W. P. Thurston}, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982; Zbl 0496.57005)]; Thurston has announced that this conjecture is true for any irreducible 3-orbifold with non-empty boundary. In this paper, the authors prove the following: If a compact 3-orbifold is built from geodesic pieces and if all of its 2-suborbifolds are good, then it is good in a strong sense, namely it is isomorphic to the quotient of a compact manifold under a finite group action. As an application, they obtain that the fundamental group of such an orbifold is residually finite, as well as a generalization to the orbifold setting of Waldhausen's theorem on the deformation of homotopy equivalences of 3- manifolds to homeomorphisms.
deformation of homotopy equivalences to homeomorphisms
scientific article; zbMATH DE number 4102086
scientific article