Separating sets in tree products, and applications

最近的进展主要研究者在拓扑结构的飞机有至关重要的依赖于意见有关的一个分离集的（笛卡尔）产品的两棵树。 也就是说，如果一个人把一棵树作为“x轴”，另一棵树作为“y轴”，那么他就可以研究什么样的（x，y）对集合会把所有对的集合分成两个或更多的部分。 现在人们发现，一个更复杂和全面的理解，这种分离集将导致进一步的进展，在连续统理论和平面拓扑学的其他重要的开放问题。 我们专注于两个老的和众所周知的问题：确定是否所有的同质平面上的（拓扑一致的，或“对称的”）空间已经被发现了。（目前只有三个紧连通的例子：圆和两个非常奇异的空间），并确定任何非分离平面连续统到自身的映射是否有不动点;也就是说，一个连续的“运动”在一个连接的平面设置没有孔必须离开一些点静止。 解决这些突出的开放性问题将大大阐明平面拓扑的性质，这是许多数学领域的基础，包括（复）分析，动力学，几何函数论和（黎曼）几何。 我们展示了一些平易近人的方向的研究和问题的分离器在树产品，并描述其直接连接到上述问题，和其他人。 借助大型而精确的计算机生成的图形，我们将仔细研究现有文献中分离集的例子，并生成新的例子和结果，这将有助于隔离和阐明平面拓扑中重要开放问题的核心困难的组合学。

Recent advances by the principal investigator in topology of the plane have depended crucially on observations pertaining to a separating set in the (Cartesian) product of two trees. That is, if one lays down a tree as the "x-axis", and another as the "y-axis", one can study what sorts of sets of (x,y) pairs will separate the set of all pairs into two or more pieces. It is now being found that a more sophisticated and comprehensive understanding of such separating sets will lead to further advances on other significant open questions in continuum theory and plane topology. We focus on two old and well-known problems: determining whether all the homogeneous (topologically uniform, or "symmetrical") spaces in the plane have already been discovered (at present there are only three compact connected examples: the circle, and two very exotic spaces), and determining whether any map of a non-separating plane continuum to itself has a fixed point; that is, whether a continuous "motion" within a connected plane set with no holes must leave some point stationary. Settling these striking open questions would substantially illuminate properties of the topology of the plane, which underlies a number of mathematical fields, including (complex) analysis, dynamics, geometric function theory, and (Riemannian) geometry. We exhibit a number of approachable directions of study and questions on separators in tree products, and describe their direct connections to the above problems, and others. With the aid of large and accurate computer-generated graphs, we will carefully study examples of separating sets in existing literature, and generate new examples and results, which will help isolate and illuminate the difficult combinatorics at the heart of significant open problems in plane topology.