Department of Mathematics
Generalized covering spaces
Andreas Zastrow
University of Gdansk, Poland
This talk will be devoted to presenting a concept of generalizing the theory of covering spaces. I and Hanspeter Fischer (Ball State University, Muncie, Indiana) have been working about it in the past years. Classical covering space has proven to be a very useful concept for semilocally simply connected spaces. It allows to present a space as a quotient of a simply connected space and gives a natural presentation for its fundamental group. Our covering spaces are constructed with the idea, to rescue these properties for a wider class of spaces. Provided that the natural homomorphism from the fundamental group into the shape group is an embedding, we obtain a simply connected universal covering space X' together with a natural projection from X' to X such that the group of p-equivariant autohomeomorphisms will be naturally isomorphic to the fundamental group of X. However, p will have weaker properties than in the classical case, but it still will have the path- and homotopy-lifting property. Our work covers also other aspects, like a universal property, intermediate covering spaces, and the weakening of the criteria if the fundamental group is countable or if the base space is first countable. These results and the particular phenomena and difficulties of this covering construction will be illustrated at a number of examples. If time suffices the talk might compare our concept with some of the other (recent and less recent) attempts of generalizing covering spaces that I am aware of.
All interested persons are welcome.