Date of Award
August 2023
Degree Type
Dissertation
Degree Name
Doctor of Philosophy
Department
Mathematics
First Advisor
Craig Guilbault
Committee Members
Jonah Gaster, Christopher Hruska, Boris Okun, Jeb Willenbring
Keywords
Boundaries, Collapsibility, Topology, Z-sets
Abstract
We extend the notion of collapsibility to non-compact complexes and prove collapsibility of locally-finite CAT(0) cube complexes. Namely, we construct such a cube complex $X$ out of nested convex compact subcomplexes $\{C_i\}_{i=0}^\infty$ with the properties that $X=\cup_{i=0}^\infty C_i$ and $C_i$ collapses to $C_{i-1}$ for all $i\ge 1$.
We then define bonding maps $r_i$ between the compacta $C_i$ and construct an inverse sequence yielding the inverse limit space $\varprojlim\{C_i,r_i\}$. This will provide a new way of Z-compactifying $X$. In particular, the process will yield a new Z-boundary, called the cubical boundary.
Recommended Citation
Gulbrandsen, Daniel L., "Collapsibility and Z-Compactifications of CAT(0) Cube Complexes" (2023). Theses and Dissertations. 3267.
https://dc.uwm.edu/etd/3267