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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.)

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2 Answers 2

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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 $r \leq n-4$.

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  • $\begingroup$ @Ebert Thanks a lot. Now I am reading this elegant paper and following the proofs. I will carefully compare this theorem with the h-cobordism theorem. $\endgroup$
    – Shinichiro Nakamura
    Commented Jun 19, 2016 at 12:01
  • $\begingroup$ @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. $\endgroup$
    – Johannes Ebert
    Commented Jun 20, 2016 at 7:54
  • $\begingroup$ @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! $\endgroup$
    – Shinichiro Nakamura
    Commented Jul 10, 2016 at 5:25
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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.

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  • $\begingroup$ @Ruberman Thank you. I was amazed that we can detect torsion number from Morse functional information. But some proofs in this paper are often sketchy and several misprints make hard to read for me. Can we get another reference that explains the Smale's theorems? $\endgroup$
    – Shinichiro Nakamura
    Commented Jun 19, 2016 at 11:53
  • $\begingroup$ There is a lot written on the subject of how to compute the homology groups of a manifold from a Morse function. Google "Morse homology" and you'll find lots of references. $\endgroup$
    – Danny Ruberman
    Commented Jun 19, 2016 at 14:18
  • $\begingroup$ @Ruberman Thank you for your comments. So, you say that theorem 6.1 in the Smale's paper can be easily deduced from Morse homology arguments, doesn't it? $\endgroup$
    – Shinichiro Nakamura
    Commented Jul 10, 2016 at 5:30
  • $\begingroup$ Well, I wouldn't say it that way; what's currently known as "Morse homology" (in the finite dimensional setting) is really an extension of Smale's work. $\endgroup$
    – Danny Ruberman
    Commented Jul 10, 2016 at 23:21

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