On the Structure of Algebraic Cobordism

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 ${\mathrm{\Omega }}_{(r)}^{\ast }(X)$ 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 $\mathbb{L}\mathrm{/}I(p,n)x$ where $\mathrm{d}\mathrm{e}\mathrm{g}x\ge \frac{p^n-1}{p-1}$. As a corollary we prove the Syzygies Conjecture of Vishik on the existence of certain free $\mathbb{L}$-resolutions of ${\mathrm{\Omega }}^{\ast }(X)$, and show that algebraic cobordism of a smooth surface can be described in terms of $K_0$ together with a topological filtration.