On Z-compactifiability of manifolds

Abstract

In 1976, Chapman and Siebenmann established necessary and sufficient conditions for $\mathbb{Z}$-compactifying Hilbert cube manifolds. Although the corresponding conditions are known to be necessary for a manifold $M^n$ to admit a $\mathbb{Z}$-compactification, it remains open whether they are sufficient. Guilbault and the author proved that they are sufficient for $M^n\times I$, when $n\ge 5$. 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 $n\ge 6$ there is a connected one-ended open PL $n$-manifold which is $\mathbb{Z}$-compactifiable but not pseudo-collarable. Finally, we discuss the additional control needed for applications to universal covers of closed aspherical manifolds.

31 pages, no figure. Substantially revised version. The destabilization theorem has been corrected by adding the necessary product-compatibility hypothesis, and the section on controlled Mather-Thurston theory has been removed. New controlled criteria and high-dimensional boundaryless examples have been added