Oriented cohomology, Borel-Moore homology and algebraic cobordism

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 $MGL'_{*,*}$ of Voevodsky's $MGL^{*,*}$-theory, and a natural map $\vartheta:\Omega_*\to MGL'_{2*,*}$, where $\Omega_*$ is the algebraic cobordism theory defined by Levine-Morel. We conjecture that $\vartheta$ is an isomorphism and describe a program for proving this conjecture.