Boundaries and End Structures for Non-compact Spaces and Groups

University Of Wisconsin-Milwaukee, Milwaukee WI

Investigators

Abstract

DMS-0072786 - Craig R. Guilbault For the sake of simplicity and convenience, topologists often focus on compact spaces. Nevertheless, non-compact spaces---with all of their associated complications---are frequently unavoidable. For example, the complement of a compact subset of a closed manifold is not compact. Hence, a topologist studying embeddings is quickly forced to consider non-compact manifolds. In fact, much of the foundational work in the area of non-compact manifold topology was motivated by questions about embeddings. While non-compact manifolds and complexes continue to play a central role in topology, an important new source of examples has entered the picture. Increasingly, the examples of widespread interest arise as covering spaces of compact manifolds and cell complexes. In fact, progress on some of the most fundamental questions in geometric topology and geometric group theory demand a better understanding of these types of spaces---most of which are non-compact. With these examples in mind, this project aims to generalize and expand our understanding of non-compact manifolds and cell complexes. Of particular interest are Z-compactifications, structure theorems for ends of non-compact manifolds, and existence and uniqueness questions for boundaries of groups. Many objects encountered in geometry and topology---like the real line and the Euclidean plane---are "unbounded" or "open" in nature. These spaces are fundamentally different from bounded or "compact" spaces, such as a line segment, a circle, or the surface of the earth. The extra flexibility permitted by unboundedness opens the door to exotic behavior "near infinity". There are many examples of unbounded spaces that---although they are constructed from simple bounded pieces---have quite complicated geometry and topology near infinity. One strategy for investigating such a space is to "compactify" it by adding a boundary at infinity---much as an artist mentally creates a horizon of vanishing points to help him draw in perspective. When this is possible, properties of the boundary can reveal important characteristics of the original space. In some sense, the boundary represents a crystallization of the "topology at infinity". Another strategy for studying an unbounded space involves breaking it into a manageable collection of bounded pieces. Ideally, one would like to choose pieces that capture key properties of the space, thus allowing one to study the non-compact space---one compact piece at a time.