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.

For a proper map with fixed complex normal orientation, apply the Thom isomorphism in complex cobordism, pull back by the collapse of a tubular neighborhood of a graph embedding, and desuspend the auxiliary Euclidean coordinates. The degree increase is the relative real codimension. Changing the orientation can change this map.

Restricting the tubular-neighborhood collapse of an embedding back to its source is the normal bundle's zero section. The Thom isomorphism in complex cobordism therefore makes pullback after the Gysin map multiplication by the Euler class in a generalized cohomology theory. For an actual complex normal bundle this is its top Chern class in complex cobordism.