A1-Homotopy Theory and Motives

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Voevodsky9901219The proposer will continue to construct the homotopy theory ofalgebraic varieties. As of today we know how to define the stable andunstable homotopy categories for algebraic varieties and thefundamental cohomology theories such as the ordinary (=motivic)cohomology, K-theory and cobordisms. The analogs of other basic ingredients of theordinary homotopy theory, such as the Steenrod operations in ordinarycohomology, are also partly understood. During this stage of the project the main attack will be made on two other ingredients which are so far missing; these are the duality theory and the recognition principle for T-loop spaces.Homotopy theory of algebraic varieties is an approach to the study ofalgebraic varieties (= systems of algebraic equations) based on ananalogy. This analogy exists between the abstract categoricalproperties of the affine line (= the system of equations with onevariable and no equations) and the toplogical space [0,1] called theunit interval. Using it one can translate methods of homotopy theory from topological spaces to algebraic varieties which gives new tools to deal with problems in algebraic geometry and number theory.