Mathematical Sciences: Classical Homotopy Theory

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9400587 Cohen F.R. Cohen intends to consider problems in classical homotopy theory and offshoots to other subjects involving cohomology of discrete groups, certain moduli spaces, and a further study of the Whitehead square. One of the main projects is to extend his recent partial solution of M.G. Barratt's finite exponent conjecture to the general case. He intends to continue an analysis of the groups appearing in this work together with their connections with other parts of mathematics involving classical homotopy theory and their connections to certain moduli spaces. He will continue his study of the Whitehead square involving an interplay between Cayley-Dickson algebras and spaces of rational functions. F.R. Cohen and S. Gitler will continue joint work in a study of characteristic classes for spaces of embeddings which are implicit in their earlier work and which are related to invariants of knots defined by Vassiliev. This project is a study of how large dimensional spheres can be wrapped around smaller dimensional objects at least up to continuous deformations. This type of geometric problem occurs ubiquitously in mathematics and other related areas such as physics. Some of the basic structure is given by the geometric and algebraic consequences of particles which move on a surface and which are parametrized by a time coordinate. At various times, the particles may return to their original position. The algebraic analogue of this geometric property is known as torsion. Bounds on the growth of torsion for various naturally occurring spaces which are analogues of surfaces is one of the main subjects of this project. One goal of this work is an attack on an outstanding problem in the subject known as Barratt's finite exponent conjecture. A second and unexpected offshoot is that this work overlaps with work of some others who are interested in physics. ***