Realization theorems for end obstructions. (English) Zbl 1160.57019

Vajiac, Bogdan

Homology Homotopy Appl. 10, No. 2, 1-12 (2008).

When is an open manifold the interior of a compact manifold with boundary? Siebenmann answered this question (for high dimensional manifolds) in his thesis [L. C. Siebenmann, Obstructions to finding a boundary for an open manifold, Ph.D. Thesis, Princeton Univ, (1965)], F. Quinn [Ann. Math. (2) 110, 275–331 (1979; Zbl 0394.57022)] provided an important controlled version, and F. Connolly and B. Vajiac [Invent. Math. 135, No. 3, 519–543 (1999; Zbl 0930.57025)] obtained stratified versions. In particular, given a manifold stratified space \(X\), there is a controlled Quinn-type obstruction whose vanishing implies that \(X\) is the interior of a compact manifold stratified space. The current paper provides examples realizing the obstructions. Moreover, an interesting example is given of a manifold stratified space with a tame end that is not the increasing union of compact manifold stratified spaces with bicollared boundaries.

Reviewer: Bruce Hughes (Nashville)

MSC:

57N80	Stratifications in topological manifolds
57N40	Neighborhoods of submanifolds
57Q10	Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57Q40	Regular neighborhoods in PL-topology
57S17	Finite transformation groups

Keywords:

stratified space; Siebenmann’s thesis; end obstructions; controlled topology

Citations:

Zbl 0394.57022; Zbl 0930.57025