Abstract
© 2019 Societe Mathematique de France. Tous droits reserves - We describe additive (unstable) operations from a theory A* obtained from the Levine-Morel algebraic cobordism by change of coefficients to any oriented cohomology theory B* (over a field of characteristic zero). We prove that there is 1-to-1 correspondence between operations An ! Bm and families of homomorphisms An ((P°°) x r ) ! B ™ ((P 1 ) X 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 ®Z Q-linear combinations of Landweber-Novikov operations which take integral values on the products of projective spaces. Furthermore, the stable operations are precisely the L-linear combinations of the 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 in¬ tegral Adams operations in algebraic cobordism, and all theories obtained from it by change of coef¬ ficients, extending the classical Adams operations in algebraic K-theory. We also construct symmetric operations and Steenrod operations (a la T. tom Dieck) in algebraic cobordism for all primes. (Only symmetric operations for the prime 2 were previously known to exist.) Finally, we prove the Riemann-Roch Theorem for additive operations which extends the multiplicative case done in [18].
Highlights
In the current article we study operations between oriented cohomology theories
These operations can be combined into a Total one which is a ”formal half” of the ”negative part” of the Total Steenrod operation in Algebraic Cobordism - see 6.4
Following D.Quillen ([22]), I.Panin-A.Smirnov ([20, Definition 3.1.1]), and M.Levine-F.Morel ([14, Definition 1.1.2]) we introduce the notion of an oriented cohomology theory on Smk
Summary
In the current article we study operations between oriented cohomology theories (over a field of characteristic zero). We describe all additive (unstable) operations in the Levine-Morel algebraic cobordism These appears to be exactly those L ⊗Z Q-linear combinations (infinite, in general) of the Landweber-Novikov operations which take ”integral” values on Ω∗((P∞)×r), for all r. We get a complete description of multiplicative operations from a free theory (in the sense of Levine-Morel) to any other theory in terms of formal group laws It is given by Theorem 6.9: Theorem 1.3 Let A∗ be a free theory, and B∗ be any oriented cohomology theory. We are able to extend a result of Panin-Smirnov and Levine-Morel on multiplicative operations Ω∗ → B∗ (see Theorem 3.7) This is done in Theorem 6.10: Theorem 1.4 Let B∗ be an oriented cohomology theory.
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