Theory of Z_n – Structures

Mr. Joe Paulson
Graduate Student
University of Wisconsin-Milwaukee

In this defense, we discuss the boundaries of Type F_n groups; those being groups whose K(G,1) complex has a finite n-skeleton. The boundaries we develop extend the notion of Z-boundaries to what we call Z_n-boundaries. This extension centers around groups no longer acting geometrically on contractible spaces, but instead n-connected spaces. Immediately this means the major theorems of “Boundary Swapping” and “Shape Equivalence of Z-Boundaries” will need revision, but a more subtle point to be discussed is that the category of spaces must also be generalized.

After discussing the foundation work for a theory of Z_n-boundaries, we end with an exploration how these new structures can be related to other well-known compactifications such as the one-point compactification, end-point compactification, and Z-compactifications.

Advisor:
Craig Guilbault

Committee Members:
Boris Okun, Chris Hruska, Jonah Gaster, and Pamela Harris