 Complex orientation of a smooth map

Represent a map $f:X\to Y$ by a smooth map, embed its graph into $Y\times {\mathbb{R}}^N$, and specify a stable complex structure on the normal bundle of that embedding, with the standard framing on the auxiliary real coordinates. Stabilization and homotopy identify equivalent choices. The auxiliary dimension is chosen to make the normal rank even; odd relative dimensions use an extra real suspension coordinate. The resulting oriented stable normal data defines the Gysin map in complex cobordism.