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. Then, my question is the following :

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

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

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.)