Homotopy theory, loop spaces,group cohomology, and configuration spaces

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Frederick R. CohenDMS-0072173The projects outlined in this proposal are directed toward the interplay between homotopy theory, group theory, and the topology of function spaces. Artin's braid group plays a central role. The main projects are as follows: (1) continuation of a program to finish Barratt's finite exponentconjecture, (2) computations of the group cohomology for certain discrete groups with varying choices of representations, (3) a further investigation of the connections between thecohomology of function spaces, group cohomology, as wellas the connections with the topology of function spaces, configuration spaces, and hyperplane arrangements, (4) an analysis of the overlap of properties of homotopy groups with other structures such as the Lie algebra attached to the descending central series for Artin's pure braid group, and the Lie algebra obtained from the higher homotopy groups of configuration spaces, and (5) a detailed analysis of certain groups of coalgebra morphisms as well as their connections to braid groups, and homotopy theory.The main goals of the projects here involve geometry of circles moving through space. These circles can be thought of as the motions of planetary objects in orbit around each other. The interplay between the geometry of these orbits occurs in mathematics as well as in mathematical physics. Precise information concerning these orbits has provided fruitful mathematical applications to several subjects. The ultimate goal of this project is to measure different quantities in the subject as well as analyzing concrete useful answers, and computations.