TITLE: Generalizations of the handle trading techniques
QUESTION [7 upvotes]: 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.)

REPLY [8 votes]: Yes; see Smale, On the structure of manifolds (Amer. J. Math. 84 1962 387–399) where it's shown that in high dimensions, you can eliminate handles under various connectivity assumptions. The h-cobordism theorem is a special case. For index greater than 1, one usually doesn't do handle-trading, because you can in fact just do handle cancellation, which is simpler. The reason for doing handle trading for 1-handles (and dually, for (n-1)-handles) instead of cancellations is to avoid tricky issues related to presentations of the fundamental group, related to the Andrews-Curtis conjecture.

REPLY [6 votes]: 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 $r \leq n-4$.