Topological Applications of Wadge Theory

With this research project, we intend to obtain topological results by using the tools of set theory. The concept of homogeneity is crucial to our investigations: a space is homogeneous if all of its points look alike. Important examples of homogeneous spaces are the topological groups: these are spaces in which the homogeneity is guaranteed by an algebraic operation like addition or multiplication. An even stronger example is given by the so-called filters: in this case, we also have a strong combinatorial structure. Our main goal is to generalize results of van Engelen from Borel spaces (these are spaces of limited complexity) to arbitrary spaces. More specifically, we plan to obtain a classification of the zero- dimensional homogeneous spaces, of the zero-dimensional topological groups, and of filters up to homeomorphism. (A space is zero-dimensional if it has a lot of gaps. Two spaces are homeomorphic if they are essentially the same space, from the point of view of topology.) The price to pay to obtain these very general results is that we will need to assume a very strong axiom, known as AD (Axiom of Determinacy), under which the universe of sets is known to be extremely regular. In fact, our main tool will be Wadge theory (which gives a deep analysis of the topological complexity of the zero-dimensional spaces), and this theory only behaves well under AD. We hope and expect that the work done in the recent joint articles with R. Carroy and S. Müller will significantly increase our chances of success.