Searching for an unabridged proof of "The Basic Theorem of Morse Theory" - MathOverflow

Searching for an unabridged proof of "The Basic Theorem of Morse Theory"

2011-07-13T17:51:47Z

Steven Smale labels the following statement "The Basic Theorem of Morse Theory" in A Survey of some Recent Developments in Differential Topology:

Let f be a $C^\infty$ function on a closed manifold with no critical points on $f^{-1}[-\epsilon,\epsilon]$ except k nondegenerate ones on $f^{-1}(0)$, all of index $s$. Then $f^{-1}[-\infty,\epsilon]$ is diffeomorphic to $X(f^{-1}[-\infty,-\epsilon];f_1,\ldots,f_k;s)$ (for suitable f_i.

Here $X(M;f;s)$ for $f\colon\,(\partial D^s)\times D^{n-s}\to M$ is M with an s-handle attached by f.
Where can I find a complete proof of this theorem, with all the t's crossed and i's dotted? Textbooks (Milnor, Matsumoto) only seem to prove homology/homotopy versions of the above statement, usually with substantial steps to be filled in by the reader. I nosed around some old papers for a few hours, (surely Smale himself proved it somewhere!) but to no avail. If I were to continue to search, no doubt I could eventually turn it up (there are a finite number of differential topology papers written 1958-1962, which is when I assume it was proven), but because I think that this question might be of wider interest, and to save me a lot of time, I'd like to ask:

Where can I find a complete unabridged proof of "The Basic Theorem of Morse Theory"? (in fact I care only about low dimensions) What is the original paper, and is there a textbook exposition of it anywhere?

Answer by John Klein for Searching for an unabridged proof of "The Basic Theorem of Morse Theory"

2011-07-13T18:08:11Z

My recollection is that Milnor's proof gives exactly what you are asking. In fact, see the remark on the bottom of page 17 of his book.

Answer by Johannes Ebert for Searching for an unabridged proof of "The Basic Theorem of Morse Theory"

2011-07-13T18:09:50Z

Kosinski, ''Differential manifolds'', Chapter VII, section 2. He gives a detailed proof in the case of just one critical point.

Answer by Pietro Majer for Searching for an unabridged proof of "The Basic Theorem of Morse Theory"

2011-07-13T21:37:33Z

R.Palais, Morse theory on Hilbert manifolds (main Theorem of §12). As you will see, in the infinite dimensional setting the construction looses nothing in clearness.

Answer by Giuseppe for Searching for an unabridged proof of "The Basic Theorem of Morse Theory"

2011-07-14T13:16:01Z

Dear Daniel Moskovich, I am answering just the last part of your question. As you said this could be useful to many others beyond the original poster, so I report my experience hoping to be useful.

The only textbook on differentiable manifolds including a proof of the basic theorem in Morse Theory, that until now I have met, is "Differentiable manifolds, Second Edition" by Lawrence Conlon.

His presentation of Morse Theory is distributed on sections 2.9.B, 3.10, and 4.2, and is closely inspired by Milnor's book.
In a certain way it requires the active cooperation of the reader by completing just some minor details, but at the end this work is doubly rewarding, it renforces your previous knowledge and assures that you grasp the content of basic morse theory.

Edit: I have found that Conlon leaves apart just to recognize that a certain manifold is a $\lambda$-handle, and for this result he refers to S.Smale "Generalized Poincarè's conjecture in dimensions greater than four"