Gysin maps in oriented theories

We prove that Gysin maps (transfer maps along closed embeddings of smooth varieties) in an oriented cohomology theory are compatible with compositions of closed embeddings. As opposed to the proof given by Panin, we do not use characteristic classes of vector bundles. Our proof is more straightforward. It is based on the properties of Thom isomorphisms and the deformation to the normal cone isomorphisms. We also provide a simpler proof of the additivity property of Gysin maps based on the use of Gysin maps with support.