Stable compactifications of polyhedra.
Steven C. Ferry · The Michigan Mathematical Journal · 2000
We prove that if X is a locally finite n-dimensional polyhedron such that X Q admits a Z-compactification, then X I 2n+5 also admits a Z-compactification. Our argument relies on an extension of Dierker's Lemma [6], [2] which says that if P p and Q q are locally finite polyhedra, p 3 and c: P! Q is a PL surjection with contractible point-inverses, then Q I 2p+1 collapses to P. See Proposition 1.5 for details, including control on the collapse. In the last section, we give an example of a uniformly contractible manifold with bounded geometry which does not satisfy Chapman-Siebenmann's tameness condition at infinity and which therefore does not admit a Z-compactification.