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Symmetric operations in algebraic cobordism. (English) Zbl 1129.14034

The author constructs certain new, so called, symmetric operations in algebraic cobrodism theory \(\Omega\) introduced by Levine and Morel. These operations are functorial, i.e., commute with pull-backs and behave in a special way (similar to the Riemann-Roch theorem) with respect to regular push-forwards. The basic property of symmetric operations is that they are trivial on the classes of embeddings and, hence, provide a natural obstruction for the cobordism class to be represented by an embedding. Another important property is that they permit to work with cobordism and algebraic cycles preserving the mod-2 torsion effects which is not possible with the usual Landweber-Novikov or Steenrod operations.
As a demonstration of these features the author provides a short and elegant computation of the algebraic cobordism of a Pfister quadric. He also obtaines several results about rationality of cycles under function field extensions. The last section of the paper can be viewed as a useful technical complement to the original book by Levine and Morel. Namely, the author provides and proves the formula of Quillen expressing the class of the projective bundle in the cobordism ring of the base, the formula for the class of the map of degree two, Excess Intersection Formula, and various formulas related to the blow-up morphism.

MSC:

14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
14C25 Algebraic cycles
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