Timeline for Searching for an unabridged proof of "The Basic Theorem of Morse Theory"
Current License: CC BY-SA 3.0
9 events
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Jul 15, 2011 at 19:06 | comment | added | agt | Dear Daniel Moskovich, it has been a pleasure to dialogue with you. | |
Jul 15, 2011 at 18:19 | comment | added | Daniel Moskovich | Thanks you for your answer. I'm probably just missing something, but I can't find the relevant details written down there at these critical steps. So (although I might be wrong) it looks to me like Smale's proof is a "sketch proof", which is all it claims to be. | |
Jul 15, 2011 at 4:49 | history | edited | agt | CC BY-SA 3.0 |
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Jul 14, 2011 at 21:18 | comment | added | agt | @Daniel Moskovich: For your first point of interest look at the sketched proof of theorem 6.2(that is your original statement), so $f^{-1}([-\infty,\varepsilon])$ is the union of $f^{-1}([-\infty,-\varepsilon])$ and of $f^{-1}([-\varepsilon,\varepsilon])$ along their common boundary $f^{-1}(\varepsilon)$, and $f^{-1}([-\varepsilon,\varepsilon])$ is diffeomorfic to $D^\lambda\times D^{n-\lambda}$, So $f^{-1}([-\infty,\varepsilon])$ is already an handlebody. For your second point of interest, starting §1, Smale says: the smooth structure obtained straightening the angles is unique up to diffeo. | |
Jul 14, 2011 at 19:56 | comment | added | Daniel Moskovich | Thanks! But I'm still having difficulty understanding. The corners I am concerned about are when you attach the cell, especially if there are many handles (one is "easy"). So what I'm looking for is where he shows that "what you contract the saddle to" is diffeomorphic to a handle $D^s\times D^{n-s}$ (smoothed somehow), and that this diffeomorphism extends over the rest of the manifold. Could you give a page reference for this step? | |
Jul 14, 2011 at 19:31 | comment | added | agt | Dear Daniel Moskovich, it seems to me that Smale says that the proof of theorem 5.1 is only sketched because its proof closely follows that of the Handlebody Theorem 1.2. Sections 2,3, and 4 are devoted to prove theorem 1.2. About straightening the angle along the corners, on page 396 in the first paragraph of §1, Smale says that he refers for such a procedure to Milnor[10] "Differentiable manifolds which are homotopy sheres". | |
Jul 14, 2011 at 19:07 | comment | added | Daniel Moskovich | Do you know where in Smale? (I was unsuccessful finding it) On page 403, there seems only to be a "proof sketch"... also, I didn't understand how corners were being treated. I mean, somehow the "idea" is in Smale, but where is the proof? | |
Jul 14, 2011 at 13:35 | history | edited | agt | CC BY-SA 3.0 |
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Jul 14, 2011 at 13:16 | history | answered | agt | CC BY-SA 3.0 |