As Theorem 8.1 in "*Lectures on the h-cobordism theorem* (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no 0,1-handles, which sometimes we call this techniques *handle trading* (we actually did trading all 1-handles with the same number of 3-handles). Then, my question is the following :


**Can we generalize handle trading techniques (for instance, trading all 2-handles with 4-handles, 3-handles with 5-handles, …) under some higher-connectivity assumptions?**


That is all of my question. Thank you for your help.


(I have edited the question to make it clear.)



Generalizations of the handle trading techniques
You might find the paper by C.T.C Wall: Geometrical connectivity I, J. London Math. Soc. 3 (1971), p. 597-604, interesting. 
What Wall proves, entirely by handle trading, is that if $W:M\_0 \to M\_1$ is an $n$-dimensional cobordism and the inclusion $M\_0\to W$ is r-connected, then you can built W from M\_0 using only handles of index $\geq r+1$, provided that $r \leq n-4$. 


