# Generalizations of the handle trading techniques

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.)

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 $r \leq n-4$.
• @snamth: Wall's paper contains half of the proof of the h-cobordism theorem. If $W$ is an h-cobordism, use Wall's theorem to find a handlebody decomposition with only handles of two consecutive indices. Then cancel those using the Whitney trick, where you have to use that the manifolds are simply connected. Jun 20, 2016 at 7:54
• @Ebert Thank you for your helpful comments. As Wall remark in the paper, if one assume that $W$ is 1-connected, then the theorem can be deduced from a proof of the h-cobordism theorem in Milnor's book. I am completely convinced, thanks! Jul 10, 2016 at 5:25