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.)