Stable and Unstable operations in Algebraic Cobordism

Abstract: We describe additive (unstable) operations from a theory A^* obtained from Algebraic Cobordism of M.Levine-F.Morel by change of coefficients to any oriented cohomology theory B^*. We prove that there is 1-to-1 correspondence between the set of operations, and the set of transformations: A^n((P^{\infty})^{\times r}) ---> B^m((P^{\infty})^{\times r}) satisfying certain simple properties. This provides an effective tool of constructing such operations. As an application, we prove that (unstable) additive operations in Algebraic Cobordism are in 1-to-1 correspondence with the L\otimes_Z Q-linear combinations of Landweber-Novikov operations which take integral values on the products of projective spaces. On our way we obtain that stable operations there are exactly L-linear combinations of Landweber-Novikov operations. We also show that multiplicative operations A^* ---> B^* are in 1-to-1 correspondence with the morphisms of the respective formal group laws. We construct Integral (!) Adams Operations in Algebraic Cobordism, and all theories obtained from it by change of coefficients, giving classical Adams operations in the case of K_0. Finally, we construct Symmetric Operations for all primes p (these operations in Algebraic Cobordism, previously known only for p=2, are more subtle than the Landweber-Novikov operations, and have applications to rationality questions), as well as the T.tom Dieck - style Steenrod operations in Algebraic Cobordism. As a bi-product of the proof of our main theorem we get the Riemann-Roch Theorem for additive (unstable) operations.