Separating sets in tree products, and applications

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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.