The principal investigators, James F. Davis and Kent E. Orr, are pursuing several problems in low and high dimensional geometric topology and exploring connections between combinatorics and topology. The problems in geometric topology include: cases of the Borel/Novikov conjecture and applications of the conjecture to manifolds with infinite fundamental group; the classical topological knot slice problem, including geometrically interpreting old and newly constructed knot slicing obstructions via Whitney tower constructions; the problem of slicing knots in even dimensions and the underlying homotopy obstructions; and better understanding the fundamental problem of classifying topological four-manifolds within a homology type. This project also is constructing the foundations of the theory of oriented matroid bundles and their characteristic classes. This requires deep interactions between two fields, topology and combinatorics and will be a step towards the understanding of a new category of manifolds, the combinatorial differential manifolds of Gelfand-MacPherson. All problems in geometric topology have their roots in the understanding of manifolds. An n-dimensional manifold is a set of points which is locally modeled on an n-dimensional linear space. For instance, a surface is a locally 2-dimensional linear space. Space-time is locally 4-dimensional, but its global properties are harder to grasp. The principal problem of geometric topology is the classification of manifolds. Two manifolds are the same if there is a continuous function with a continuous inverse between the manifolds. This formalizes the notion of rubber sheet topology often discussed in popular science magazines. Another natural aspect of manifold theory is the question of how manifolds can sit within other manifolds, i.e., knot theory. One pathway to understanding manifolds is through the fundamental group and homology theory, which provides a transition from geometry to algebra. Another pathway is to approximate geometry by finite structures, which provides a transition from geometry to combinatorics. The interaction of these fields of mathematics reveals techniques and perspectives that would otherwise be invisible. ***
