Equivariant, Almost-Arborescent Representations of Open simply-Connected 3-Manifolds; A Finiteness result

When one extends the (almost) collapsible pseudo-spine representation theorem for homotopy 3-spheres [Po3] to open simply connected 3-manifolds V 3,new phenomena appear: at the source of the representation, the set of double points is, generally speaking, no longer closed. We show that at the cost of replacing V3 by Vh3 = {V3 with very many holes }, we can always find representations X2 →f V3 with X2 locally finite and almost-arborescent, with Ψ (f) = Φ(f), with the open regular neighbourhood (the only one which is well-defined here) Nbd(f X2) = Vh3 and such that on any precompact tight transversal to the set of double lines, we have only FINITELY many limit points (of the set of double poi…