This research project will focus on closed, continuous functions defined on manifolds in which all point preimages are closed, connected manifolds of a given dimension. The emphasis will be to analyze the image spaces, to develop conditions implying they have nice local behavior (causing them to be, say, ANR's or, better, generalized manifolds), and finally to assess the structure of the possible source domains. The new research tools put to work on this will be algebraic, the most important being sheaf theoretic ones. The drawing together of different branches of topology in recent years is strikingly illustrated by this project.
