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Theorem 1 of Milnor's paper "A procedure for killing homotopy groups of differentiable manifolds" states that two manifolds are in the same cobordism class if and only if they can be obtained from each other by a sequence of surgeries (meaning they are $\chi$-equivalent, following Milnor's terminology).

In his proof that cobordism implies $\chi$-equivalence, he utilizes a Morse function on the cobordism manifold, although not explicitly. Is there a reference containing a more modern reformulation of this proof via Morse functions? I would also like to avoid introducing the language of attaching handles if possible.

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    $\begingroup$ I would be very surprised to learn that the method of attaching handles has been usurped by some "more modern reformulation". $\endgroup$
    – Lee Mosher
    Feb 3, 2022 at 1:34

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If you want to avoid the language of attaching handles there is the treatment in Milnor's Lectures on the h-cobordism theorem. He proves that a cobordism with a Morse function with only one critical point (an `elementary cobordism') gives rise to a surgery, which is the statement you're looking for. To a modern reader, the handles are there even if that terminology isn't used.

I remember hearing as a grad student that Milnor wrote the lectures in that way because Morse didn't like handles (which was the way that Smale described his argument) or was worried about doing handle attachments smoothly. So (according to this possibly urban legend) Milnor phrased things in a way that Morse (who again allegedly attended the lectures) the statements were all about eliminating critical points of a Morse function rather than cancelling handles and so on. I would be interested if anyone can provide any evidence for this story.

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    $\begingroup$ I've certainly read about Morse's attempt to effectively redo the h-cobordism theorem at the level of Morse functions and homotopies of Morse functions, rather than using handle attachments. I would imagine I read this in the Kosinski book, or perhaps in Dieudonne's book. Will take me a few days to find those passages. $\endgroup$
    – Ryan Budney
    Feb 3, 2022 at 5:02
  • $\begingroup$ Is there a proof for such a statement extended to all cobordisms, not just elementary ones? $\endgroup$
    – Luke McEvoy
    Feb 3, 2022 at 12:50
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    $\begingroup$ The statement is that any cobordism admits a Morse function such that the critical points appear at different levels. That automatically decomposes it into a series of elementary cobordisms, each of which is described as in that chapter of Milnor's book. Hence your sequence of surgeries. $\endgroup$
    – Danny Ruberman
    Feb 3, 2022 at 13:05

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