2023/12/05 by Gu, Shijie
#FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2312.02527
In 1976, Chapman and Siebenmann \citeCS76 established necessary and sufficient conditions for Z-compactifying Hilbert cube manifolds. While these conditions are known to be necessary for a manifold Mn to admit a Z-compactification, it remains an open question whether these conditions are also sufficient. Guilbault and the author \cite[Thm. 1.2]GG20 proved that these conditions are sufficient for the product Mn × [-2,2] (n≥ 5) to be Z-compactifiable. We further explore this topic by introducing additional conditions such that a Z-compactification of Mn × [-2,2] indeed implies a Z-compactification of Mn, thus partially resolving the open question. As applications, it is shown that there exist infinitely many non-pseudo-collarable 4-manifolds which are Z-compactifiable; however, pseudo-collarable manifolds with compact boundary of dimension at least six are Z-compactifiable. Furthermore, we investigate the connection between Z-compactifiability with the topological rigidity of aspherical manifolds. We also construct a noncompact one-sided s-cobordism (W,V,V∗) satisfying controlled Mather-Thurston theorems, where V is Z-compactifiable, whereas V∗ may not be.