Two investigators are involved in this project. James F. Davis' main emphasis is the use of surgery theory to study rational information in manifold theory. Surgery theory is the main tool for classification of manifolds of dimension greater than 3, although complexities in homotopy and integral representation theory make it efficacious only in special cases. However, if one works over the rational numbers, complete results are available for manifolds with finite fundamental group. Davis will attempt extensions of these results to manifolds with infinite fundamental group, and to spaces more general than manifolds, but which still possess some form of duality. A second and quite different direction of Davis' research is knot theory. He will study the homology of covers of the 3-sphere branched over a knot, the set of periods of a knot, and their relationship with algebraic number theory. In related work, Kent E. Orr will continue to study linking and knotting phenomena in codimension two, as well as investigate the implications of the recent Cochran-Orr work to Casson-Gordon invariants and four-dimensional surgery. Among other things, he hopes to answer the following two questions about the link slice problem: Are all even-dimensional links slice? Do the presently known homotopy theoretic linking invariants of links in the 3-sphere (such as the transfinite invariants of Levine and Orr and the Le Dimet invariant) reduce to classically defined invariants such as Milnor's invariants and Massy products? The details of these parts vary, but all are concerned with reducing geometric information to a subject for calculation or to perfecting the algebraic machinery used for the calculations. The nature of the geometric information involved is the crux of the difficulty. While questions about lengths, areas, angles, volumes, and so forth virtually cry out to be reduced to calculations, it is far different with what are known as topological properties of geometric objects. These are properties such as connectedness (being all in one piece), knottedness, having no holes, and so forth. All systematic study of such properties, for example, how to tell whether two geometric objects really differ in respect to one of these properties or are only superficially different, or how to classify the variety of differences that can occur, all these have only truly been comprehended and mastered when they have been reduced to matters of calculation, and two of the principal tools for this are homotopy theory and surgery theory.
