Symmetric operations in algebraic cobordism

In this article we describe certain new cohomological operations in algebraic cobordisms. These operations give the natural obstructions for the cobordism class to be represented by the embedding. Also, they permit to work with algebraic cobordisms and Chow groups in a more subtle way than the Landweber–Novikov operations (related to 2-torsion effects). We describe applications to the computation of the algebraic cobordisms of a Pfister quadrics and to the problem of rationality of cycles.