We define and study a new oriented cohomology theory MGL*,*sst (-) defined on smooth complex varieties, called semi-topological cobordism. The construction combines the semitopological construction due to Friedlander-Walker together with motivic stable homotopy theory for smooth varieties. The semi-topological cobordism interpolates between the algebraic cobordism MGL*,* defined by Voevodsky in the motivic stable homotopy category, and the complex cobordism of the underlying analytic variety MU *(-). The algebraic cobordism and semi-topological cobordism agree with finite coefficients. On the other hand the semi-topological cobordism and the topological complex cobordism agree integrally for a point as well as certain classes of varieties such as smooth cellular varieties or smooth projective toric varieties.
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