I am currently learning Morse theory. Having read Milnors "Morse Theory" I am now studying his "Lectures on the h-cobordism theorem". I have also read parts of "Lectures on Morse Homology" by Banyaga and Hurtubise as well as several texts that I could find online.
As far as I understand it, Milnor constructs the Morse complex on his way to the proof of the h-cobordism theorem.
My question is: If I have already established Morse homology and proven its equivalence to simplicial homology, is there any easier/shorter proof of the h-cobordism theorem basing on these results?
My question is somewhat vague and may resolve when I read further, but maybe someone who already has the "big picture" can help me with this.
The heart of the proof of Smale's h-cobordism theorem is handle rearrangement, there is no shortcut or way around it.