Mathematical Sciences: Applications of Algebraic Topology to Fixed Point Theory, Dynamics, and Cohomology of Groups

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9401073 Geoghegan Professor Geoghegan will investigate problems at the interface of algebra and topology, involving fixed point theory, dynamics and group theory. In collaboration with the Canadian topologist A. Nicas, he has developed a new "one-parameter fixed point theory." From this come trace formulae for Fuller index-type invariants of non-singular flows, and new K_1-invariants and "non-commutative" zeta functions for suspension flows. The extension of the latter to non-suspension flows is a goal of the present project. Their "first order Euler characteristic" is a new invariant in topology and group theory having properties analogous to the classical ("zeroth order") Euler characteristic. They will continue to compute this in specific cases and develop a full theory. The proposal also covers Professor Geoghegan's interests in geometric group theory, with particular reference to local properties of K(G,1) complexes which imply that G is a duality group, in the sense of Bieri and Eckmann. He seeks a criterion having the same relationship to duality groups as the manifold property has to Poincare duality groups. This project is about the connections between topology, which is a kind of fundamental geometry, and group theory, which concerns the formal study of symmetries. The layperson is not accustomed to separating the "group" of symmetries possessed by an object from the object itself, but group theory does just that. It transfers the symmetries from where they first arose to new mathematical objects (where they again appear as symmetries but in a different way), thus revealing subtle new features of the group. One aspect of this general philosophy, sometimes called Geometric Group Theory, reaches into the depths of Algebraic Topology, with its enormous wealth of methods and examples, to find new and revealing settings for the group being studied. In the present instance, the methods of time-dependent fixed point theory are particular ly relevant. The investigator and his collaborator have developed a new aspect of this part of topology and are using it to detect new invariants of certain kinds of groups. While application outside mathematics is not the driving force, they have had success in applying their work to Dynamics, specifically to the behavior of closed orbits of flows in certain situations, dynamics now being considered seriously applicable mathematics. Furthermore, the investigator has participated in a collaboration with an applied mathematician and an electrical engineer to use some of his methods in the theory of electrical circuits. ***