Award: DMS 1200284, Principal Investigator: Roy Joshua
The PI, together with Gunnar Carlsson at Stanford, has been involved in a research program over the last three years with the goal of constructing and setting up a motivic variant of the Becker-Gottlieb transfer and using this to study several problems in motivic cohomology and algebraic K-theory with finite coefficients. This collaborative work over the last three years has already resulted in two preprints, where such a transfer has already been constructed and a basic framework for subsequent work established. The goal of the present research proposal is to explore applications of this transfer to motivic cohomology and algebraic K-theory with finite coefficients with several such applications expected. The corresponding classical notions in algebraic topology have played a key role in settling important problems like the Adams? conjecture, where the use of the transfer provided a sophisticated version of the splitting principle. A leading theme of the proposed research is that other more sophisticated applications of the same principle accounts for several of the deeper results in algebraic K-theory: for example, the reduction of Thomason's well-known result on cohomological descent for mod-l algebraic K-theory with the Bott-element inverted to Hilbert's theorem 90, and the reduction of rigidity properties of mod-l algebraic K-theory to Picard groups. However, in the absence of an analogue of the Becker-Gottlieb transfer, these reductions have been rather long and difficult, requiring the development of customized elaborate machinery. There are also currently several open problems to which similar techniques seem to apply: for example, Carlsson's conjectures relating the algebraic K-theory of fields with certain equivariant K-groups associated to the Galois group of the field and also several open questions on the motivic cohomology of the classifying spaces of algebraic groups.
Motivic cohomology and motivic homotopy theory are relatively new applications of well-established techniques from algebraic topology to algebraic geometry. Classically cohomology and homotopy methods are used to distinguish between topological spaces and to analyze their properties. Algebraic geometry, on the other hand, deals with special spaces arising as zeroes of polynomial equations. It is only in recent years, after the solution of an open problem called the Milnor conjecture using homotopy methods that the relevance and usefulness of such techniques in algebraic geometry have come to light. Considering that the Becker-Gottlieb transfer is one of the most versatile tools in algebraic topology, it is reasonable to expect much progress using a variant of it in the motivic context. The importance and relevance of this project stems from these observations.
