On the Structure of Algebraic Cobordism

TL;DR

This paper explores the detailed algebraic structure of Levine-Morel's algebraic cobordism, revealing specific module decompositions, proving a conjecture on free resolutions, and connecting cobordism of surfaces to K-theory.

Contribution

It provides a detailed description of the module structure of algebraic cobordism, proves Vishik's Syzygies Conjecture, and relates cobordism of surfaces to K-theory.

Findings

Associated graded groups are unions of finitely presented modules.

Submodules have filtrations with free or cyclic factors.

Proves the Syzygies Conjecture of Vishik.

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.