Stable and Unstable Operations in Algebraic Cobordism

Vishik, Alexander

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 52 (2019) no. 3, pp. 561-630.

Nous décrivons les opérations additives (instables) d'une théorie A* obtenue par changement de coefficients à partir du cobordisme algébrique de Levine-Morel vers une théorie cohomologique orientée quelconque B* (sur un corps de caractéristique nulle). Nous établissons une correspondance bijective entre les opérations An→Bm et les familles de morphismes An((ℙ∞)×r)→Bm((ℙ∞)×r) satisfaisant certaines propriétés simples. Cela fournit une manière effective de construire de telles opérations. Comme application, nous prouvons que les opérations additives (instables) internes au cobordisme algébrique sont en correspondance bijective avec les combinaisons 𝕃⊗ℤℚ-linéaires des opérations de Landweber-Novikov. Nous montrons également que les opérations multiplicatives A*→B* sont en correspondance bijective avec les morphismes entre les lois de groupes formels respectives. Nous construisons des opérations d'Adams sans dénominateurs en cobordisme algébrique et en toute théorie obtenue à partir du cobordisme algébrique par changement de coefficients, qui étendent les opérations classiques d'Adams en K-théorie. Nous construisons également des opérations symétriques et de Steenrod (à la T. tom Dieck) en cobordisme algébrique, pour tout nombre premier. (Seules les opérations symétriques pour le nombre premier 2 étaient définies auparavant). Enfin, nous prouvons le théorème de Riemann-Roch pour les opérations additives, ce qui généralise le cas multiplicatif traité en [18].

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∞)×r)→Bm((ℙ∞)×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⊗ℤℚ-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 integral Adams operations in algebraic cobordism, and all theories obtained from it by change of coefficients, 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].

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