Comparison of cobordism theories

Abstract

Relying on results of Hopkins-Morel, we show that, for $X$ a quasi-projective variety over a field of characteristic zero, the canonical map ${\Omega }_n(X)\to MG{L}_{2n,n}^{\prime }(X)$ is an isomorphism. Here ${\Omega }_{\ast }(X)$ is the theory of algebraic cobordism defined by Levine-Morel, and $MG{L}_{\ast ,\ast }^{\prime }$ is the Borel-Moore homology version of the theory of algebraic cobordism defined via the algebraic Thom complex in the Morel-Voevodsky motivic stable homotopy category.

22 pages