Oriented cohomology, Borel-Moore homology and algebraic cobordism

Abstract

We examine various versions of oriented cohomology and Borel-Moore homology theories in algebraic geometry and put these two together in the setting of an "oriented duality theory", a generalization of Bloch-Ogus twisted duality theory. This combines and exends work of Panin and Mocanasu. We apply this to give a Borel-Moore homology version $MG{L}_{\ast ,\ast }^{\prime }$ of Voevodsky's $MG{L}^{\ast ,\ast }$-theory, and a natural map $\vartheta :{\Omega }_{\ast }\to MG{L}_{2\ast ,\ast }^{\prime }$, where ${\Omega }_{\ast }$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\vartheta$ is an isomorphism and describe a program for proving this conjecture.

47 pages. To appear in Michigan Math. J