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Full-Text Articles in Physical Sciences and Mathematics
Compactifications Of Manifolds With Boundary, Shijie Gu
Compactifications Of Manifolds With Boundary, Shijie Gu
Theses and Dissertations
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. …
Z-Structures And Semidirect Products With An Infinite Cyclic Group, Brian Walter Pietsch
Z-Structures And Semidirect Products With An Infinite Cyclic Group, Brian Walter Pietsch
Theses and Dissertations
Z-structures were originally formulated by Bestvina in order to axiomatize the properties that an ideal group boundary should have. In this dissertation, we prove that if a given group admits a Z-structure, then any semidirect product of that group with an infinite cyclic group will also admit a Z-structure. We then show how this can be applied to 3-manifold groups and strongly polycyclic groups.