Boundaries and End Structures for Non-compact Spaces and Groups

DMS-0072786 -克雷格R. Guilbault为了简单和方便起见，拓扑学家经常关注紧空间。然而，非紧致空间-以及所有与之相关的复杂性-常常是不可避免的。例如，闭流形的紧致子集的补不是紧致的。因此，研究嵌入的拓扑学家很快就被迫考虑非紧流形。事实上，非紧流形拓扑领域的许多基础工作都是由嵌入问题引起的。 当非紧流形和复形继续在拓扑学中扮演中心角色时，一个重要的新的例子来源已经出现。越来越多的广泛感兴趣的例子出现在紧流形和胞腔复形的覆盖空间中。事实上，在几何拓扑学和几何群论中一些最基本的问题上的进展需要更好地理解这些类型的空间-其中大多数是非紧的。 考虑到这些例子，这个项目旨在推广和扩展我们对非紧流形和胞腔复合体的理解。特别感兴趣的是Z-紧化、非紧流形端点的结构定理以及群边界的存在唯一性问题。几何和拓扑学中遇到的许多对象--如真实的直线和欧几里得平面--在本质上是“无界的”或“开的”。这些空间与有界或“紧凑”空间（例如直线段、圆或地球表面）有根本不同。无限性所允许的额外灵活性为“接近无穷大”的奇异行为打开了大门。有很多例子表明，尽管无界空间是由简单的有界部分构成的，但它们在无穷远处具有相当复杂的几何和拓扑结构。研究这样一个空间的一个策略是通过在无穷远处增加一个边界来“压缩”它-就像艺术家在脑海中创造一个消失点的地平线来帮助他透视画一样。如果这是可能的，边界的性质可以揭示原始空间的重要特征。在某种意义上，边界代表了“无限拓扑”的结晶。 研究无界空间的另一个策略是将其分解为可管理的有界片段集合。理想情况下，人们会选择捕捉空间关键属性的片段，从而允许人们研究非紧凑空间-每次一个紧凑片段。

DMS-0072786 - Craig R. GuilbaultFor 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 ofwhich 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 uniquenessquestions 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 aline 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---althoughthey 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.