Mathematical Sciences: Knots, Framed Manifolds, and Jet Groups

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This research project involves three different areas of topology: knot theory, manifold theory, and homology of groups. The problems considered are handled by a variety of techniques, coming from differential topology (cutting and pasting, transversality, transformation groups), homotopy theory (homotopy groups of spheres, plus construction, S-duality), and algebra (homological algebra, combinatorial group theory, symmetric forms). Part 1 is an investigation of certain classes of higher- dimensional knots. The first class consists of knots which are not determined by their complements. Only recently has it been proved that such knots do not occur in 3-space. On the other hand, the investigator has since shown that such knots do occur in at least a quarter of all higher dimensions. He intends to find inequivalent knots with the same complement in the missing dimensions. The second class consists of certain knots whose complements have cyclic fundamental group. The classification of such knots will be pursued by means of algebraic invariants associated to their complements. The third class consists of knotted higher-genus surfaces in 4-space. That this is a much richer collection than that of ordinary knotted 2-spheres will be shown by an analysis of the peripheral structure of the knot groups. Part 2 concerns a special class of closed manifolds -- those obtained by spinning lower-dimensional ones about framed submanifolds of spheres. Using a generalized Pontrjagin-Thom construction, the investigator will carry out the homotopy classification of such manifolds in certain cases. Applications to the bordism classification of homology spheres will be given. Part 3 describes an ongoing program with S. Jekel for computing the homology of the classifying space for codimension- one real analytic singular foliations. The investigator outlines the calculation of the homology of discrete groups of jets of local diffeomorphisms, and of some related nilpotent groups. In short, the investigator will apply himself on the one hand, to some very concrete geometric problems concerning higher dimensional knots and symmetries of manifolds, and on the other hand, to enlarging the scope of algebraic tools for attacking such problems.