Compactifications of Manifolds with Boundary

Abstract

This dissertation is concerned with compactifications of high-dimensional manifolds. Siebenmann's iconic 1965 dissertation \cite{Sie65} provided necessary and sufficient conditions for an open manifold $M^{m}$ ($m\geq6$) to be compactifiable by addition of a manifold boundary. His theorem extends easily to cases where $M^{m}$ is noncompact with compact boundary; however when $\partial M^{m}$ is noncompact, the situation is more complicated. The goal becomes a \textquotedblleft completion\textquotedblright\ of $M^{m}$, ie, a compact manifold $\widehat{M}^{m}$ containing a compactum $A\subseteq\partial M^{m}$ such that $\widehat{M}^{m}\backslash A\approx M^{m}$. Siebenmann did some initial work on this topic, and O'Brien \cite{O'B83} extended that work to an important special case. But, until now, a complete characterization had yet to emerge. Here we provide such a characterization. Our second main theorem involves $\mathcal{Z}$-compactifications. An important open question asks whether a well-known set of conditions laid out by Chapman and Siebenmann \cite{CS76} guarantee $\mathcal{Z}$-compactifiability for a manifold $M^{m}$. We cannot answer that question, but we do show that those conditions are satisfied if and only if $M^{m}\times\lbrack0,1]$ is $\mathcal{Z}$-compactifiable. A key ingredient in our proof is the above Manifold Completion Theorem---an application that partly explains our current interest in that topic, and also illustrates the utility of the $\pi_{1}% $-condition found in that theorem. Chapter \ref{Chapter 1} is based on joint work with Professor Craig Guilbault \cite{GG17}. At last, we obtain a complete characterization of pseudo-collarable $n$-manifolds for $n\geq 6$. This extends earlier work by Guilbault and Tinsley to allow for manifolds with noncompact boundary. In the same way that their work can be viewed as an extension of Siebenmann's dissertation that can be applied to manifolds with non-stable fundamental group at infinity, Pseudo-collarability Characterization Theorem can also be viewed as an extension of Manifold Completion Theorem in a manner that is applicable to manifolds whose fundamental group at infinity is not peripherally stable.