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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 PoincarĂ¨'s conjecture in dimensions greater than four"

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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 PoincarĂ¨'s conjecture in dimensions greater than four"

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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.