Mathematical Sciences: Topological Applications of Algebraic Cycles

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9401533 Lima-Filho The investigator will continue to explore topological properties of the groups of algebraic cycles on algebraic varieties. He has found this to be a mine of applications to topological and algebraic geometric problems. Algebraic cycles are seen as a topological group functor from the category of algebraic varieties to the category of group objects in the category of CW-complexes. This functor associates principal fibrations to closed inclusions, and their homotopy groups provide a homology theory for algebraic varieties which he wishes to understood more deeply. An intersection theory for arbitrary varieties will be developed, generalizing and simplifying an existing theory for quasiprojective varieties. On the purely topological side, the investigator's research has created equivariant infinite loop spaces using algebraic cycles, and the generalized cohomology theories thus obtained have been related with several other theories. As an outcome of this interplay between new and old theories, he has already answered long-standing conjectures about the total Chern class map and is presently computing several examples where algebraic geometry, group representation theory, and homotopy theory come together to illuminate the coefficient systems involved, the associated transfer maps, and so forth. A fundamental aspect and guiding principle in the investigator's research is its unifying character, under which diverse areas of mathematics come into play and yield multifaceted applications of his algebraic techniques. He thus obtains a highly desirable economy of thought, in the sense that a concise and unified approach to various seemingly disparate questions attains deep results about each in a short period of time. He starts by endowing basic objects in algebraic geometry, the algebraic cycles, with a topology which turns out to have remarkable properties. He then realizes that these cycles provide new invariants not only for alge braic varieties (of fundamental importance to algebraic geometers), but that they also deeply relate with ongoing mainstream research in stable homotopy theory, a major topic in algebraic topology. Other natural structures arise in this setting as he explores constructions involving his algebraic cycles. The study of cycles from an equivariant point of view thus also provides insight into various aspects of the theory of representation of finite groups and group cohomology, enhancing the usefulness of his unified approach. ***