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Stable and unstable operations in algebraic cobordism

Alexander VISHIK, Alexander

Stable and unstable operations in algebraic cobordism Thumbnail



© 2019 Société Mathématique de France. Tous droits réservés - 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 (à 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].


Alexander VISHIK, A. (2019). Stable and unstable operations in algebraic cobordism. Annales Scientifiques de l'École Normale Supérieure, 1(52), 561-630.

Journal Article Type Article
Acceptance Date Sep 28, 2017
Online Publication Date May 30, 2019
Publication Date May 30, 2019
Deposit Date Oct 3, 2017
Publicly Available Date Oct 3, 2018
Journal Annales scientifiques de l'École normale supérieure
Print ISSN 0012-9593
Electronic ISSN 0012-9593
Publisher Elsevier
Peer Reviewed Peer Reviewed
Volume 1
Issue 52
Pages 561-630
Keywords General Mathematics
Public URL
Publisher URL
Related Public URLs
Contract Date Oct 3, 2017


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