Just about all of the standard textbooks on manifold theory give proofs of weak versions of the Whitney Embedding theorem. But other then Whitney's original 1944 paper,are there any standard sources that contain a full proof of the strong version of the theorem? The only textbook I know that contains a full proof is Prasolov's Elements of Homology Theory-a wonderful book on algebraic topology I'd recommend to any student studying the subject. But does anyone know any other book sources for it? Just curious.
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It's in Adachi's "Embeddings and Immersions" which is part of the AMS translations series.
Part of the reason why you don't see it written up on its own very often is that the key idea of the proof is used for the proof of the h-cobordism theorem. So most people see the argument in the h-cobordism theorem (called "the Whitney trick") and figure out the proof of the strong embedding theorem from that.
For the Whitney Trick, the main source I think most people use is Milnor's lecture notes on the h-cobordism theorem. A Google search should quickly get you a .pdf or .djvu of the book.
I would use as a general source for Whitney's embedding theorem "Global Analysis" by D. Kahn. This covers the basics. For more far leading resources, I would hurry up to fetch a copy of Whitney original work. This is most illuminating because Whitney really suceeded in explaing (implicitly only, of course) what's behind his ideas.