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.



A question on the proof of Theorem 1 in Milnor's "Killing Homotopy Groups"
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.


