Stable and unstable operations in algebraic cobordism

Alexander VISHIK, Alexander

Abstract

© 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].

Citation

Alexander VISHIK, A. (2019). Stable and unstable operations in algebraic cobordism. Annales Scientifiques de l'École Normale Supérieure, 1(52), 561-630. https://doi.org/10.24033/asens.2393

• 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: 1873-2151
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 1
• Issue: 52
• Pages: 561-630
• DOI: https://doi.org/10.24033/asens.2393
• Keywords: General Mathematics
• Public URL: https://nottingham-repository.worktribe.com/output/884977
• Publisher URL: https://smf.emath.fr/publications/operations-stables-et-instables-en-cobordisme-algebrique
• Related Public URLs: https://www.math.ens.fr/edition/annales/