On Z-compactifiability of manifolds

In 1976, Chapman and Siebenmann \cite{CS76} established necessary and sufficient conditions for $\mathcal{Z}$-compactifying Hilbert cube manifolds. While these conditions are known to be necessary for a manifold $M^n$ to admit a $\mathcal{Z}$-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 $M^n\times [-2,2]$ $(n\ge 5)$ to be $\mathcal{Z}$-compactifiable. We further explore this topic by introducing additional conditions such that a $\mathcal{Z}$-compactification of $M^n\times [-2,2]$ indeed implies a $\mathcal{Z}$-compactification of $M^n$, thus partially resolving the open question. As applications, it is shown that there exist infinitely many non-pseudo-collarable 4-manifolds which are $\mathcal{Z}$-compactifiable; however, pseudo-collarable manifolds with compact boundary of dimension at least six are $\mathcal{Z}$-compactifiable. Furthermore, we investigate the connection between $\mathcal{Z}$-compactifiability with the topological rigidity of aspherical manifolds. We also construct a noncompact one-sided $s$-cobordism $(W,V,V^{\ast })$ satisfying controlled Mather-Thurston theorems, where $V$ is $\mathcal{Z}$-compactifiable, whereas $V^{\ast }$ may not be.