9401086 Daverman The aim of this project is to characterize (1) low-dimensional manifolds and (2) a class of nicely structured functions -approximate fibrations - defined on them. Both components involve close scrutiny of continuous functions on these objects, with severe restrictions imposed on the point preimages. The first comprises the final aspects of a far-reaching effort to characterize manifolds topologically; primarily it calls for determination of an appropriate geometric notion of general position to fulfill the role played by the Disjoint Disks Property in the characterization of high dimensional manifolds. The second arises because recent results in the piecewise linear category make it reasonable to expect discovery of widespread additional conditions under which one quickly and easily detects an approximate fibration; specifically, it seems that for many n-manifolds N , any map from an (n+k)-manifold onto a reasonable space X with each point preimage homotopy equivalent to N will be an approximate fibration. Interest in this class stems from andtrades upon computable relationships among domain, range, and typical fiber. New tools are sought for this effort, geometric ones in the first case, to distinguish genuine manifolds from pretenders, and algebraic ones in the second case, both to improve the recognition criteria and to expand the applications. The projected research activity focuses on manifolds, a foundational project in that these objects are central to so much mathematics. One aim is to characterize low-dimensional manifolds, which would crown the far-reaching effort to characterize manifolds topologically, with complete success recently achieved in dimensions greater than 4, and hopes for related success in the two remaining dimensions of interest. A second aim is to characterize certain nicely structured functions, approximate fibrations, defined on manifolds. This effort appears promising now because several resu lts from the past two years make it reasonable to expect discovery of additional, useful conditions under which one could quickly detect this class of functions. The resultant ease and variety of computations about the various objects at hand would be a significant benefit. New tools are sought for each of these efforts, geometric ones in the first case, to distinguish genuine manifolds from pretenders, and algebraic ones in the second case, to provide additional relationships among domain, range, and fiber of the functions under consideration and thereby to expand the applications. ***
