On the Structure of Algebraic Cobordism

Abstract

In this paper we investigate the structure of algebraic cobordism of Levine-Morel as a module over the Lazard ring with the action of Landweber-Novikov and symmetric operations on it. We show that the associated graded groups of algebraic cobordism with respect to the topological filtration $\Omega^*_{(r)}(X)$ are unions of finitely presented $\mathbb{L}$-modules of very specific structure. Namely, these submodules possess a filtration such that the corresponding factors are either free or isomorphic to cyclic modules $\mathbb{L}/I(p,n)x$ where $\mathrm{deg\ } x\ge \frac{p^n-1}{p-1}$. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free $\mathbb{L}$-resolutions of $\Omega^*(X)$, and show that algebraic cobordism of a smooth surface can be described in terms of $K_0$ together with a topological filtration.