On Z-compactifiability of manifolds

Abstract

In 1976, Chapman and Siebenmann established necessary and sufficient conditions for $\Z$-compactifying Hilbert cube manifolds. Although the corresponding conditions are known to be necessary for a manifold $M^n$ to admit a $\Z$-compactification, it remains open whether they are sufficient. Guilbault and the author proved that they are sufficient for $M^n\times I$, when $n\geq5$. We further explore this question by giving additional hypotheses under which the interval factor can be removed. A retraction defined near the central added set is sufficient; a product-compatible splitting also gives a collar; and an upper-semicontinuous cell-like decomposition gives a splitting-free criterion. We also answer affirmatively a question of Guilbault--Tinsley by showing that: for every $n\ge6$ there is a connected one-ended open PL $n$-manifold which is $\Z$-compactifiable but not pseudo-collarable. Finally, we discuss the additional control needed for applications to universal covers of closed aspherical manifolds.