Realization theorems for end obstructions. (English) Zbl 1160.57019

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.


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
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