Algebraic cobordism

TL;DR

This paper surveys the development of algebraic cobordism, an algebro-geometric analogue of complex cobordism, and discusses potential extensions to higher algebraic cobordism within the motivic homotopy framework.

Contribution

It reviews the construction and main results of algebraic cobordism and proposes a candidate for higher algebraic cobordism theory aligned with motivic homotopy theory.

Findings

Algebraic cobordism is constructed as an algebro-geometric analogue of complex cobordism.

Main results include the properties and applications of algebraic cobordism.

A candidate for higher algebraic cobordism is proposed, potentially matching the $MGL$ cohomology theory.

Abstract

Together with F. Morel, we have constructed in \cite{CR, Cobord1, Cobord2} a theory of {\em algebraic cobordism}, an algebro-geometric version of the topological theory of complex cobordism. In this paper, we give a survey of the construction and main results of this theory; in the final section, we propose a candidate for a theory of higher algebraic cobordism, which hopefully agrees with the cohomology theory represented by the $\P^1$-spectrum $MGL$ in the Morel-Voevodsky stable homotopy category.