Most general context for the Morse Lemmas - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T20:34:18Z http://mathoverflow.net/feeds/question/80213 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80213/most-general-context-for-the-morse-lemmas Most general context for the Morse Lemmas Daniel Moskovich 2011-11-06T14:25:37Z 2012-01-29T12:43:49Z <p>Among the foundational results in differential topology are the <a href="http://en.wikipedia.org/wiki/Morse_function#The_Morse_lemma" rel="nofollow">Morse lemmas</a>:</p> <ol> <li>Suppose that $f\colon\, M\to \mathbb{R}$ is a smooth function on a closed manifold M, that $f^{−1}[-\epsilon,\epsilon]$ is compact, and that there are no critical values between $-\epsilon$ and $\epsilon$. Then $f^{-1}(-\infty,-\epsilon]$ is diffeomorphic to $f^{-1}(-\infty,\epsilon]$.</li> <li> Let $f\colon\, M\to \mathbb{R}$ be a smooth function on a closed manifold M, 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<sub>i</sub>).<br> 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. </li> </ol> <p>In plain English, the Morse lemmas give us instructions for how to build M out of simple pieces, like a child would build a structure out of Lego blocks. The first lemma says "if f has no critical point, do nothing", while the second lemma says "if f has a critical point, glue in an appropriate handle". <br><br></p> <p>One of the things that makes me feel that I don't understand Morse's lemmas as well as I would like to is that the conditions on the source and target of the Morse function f seem unnecessarily restrictive. Maybe we'd like M to be a manifold with boundary or with corners, or a stratified space, or an infinite-dimensional something-or-other? Indeed, analogues of the Morse lemmas continue to hold (but how much CAN we relax our requirements on M?).<br><br> On the target side, what about if we want the target to be something other than $\mathbb{R}$? <a href="http://en.wikipedia.org/wiki/Circle-valued_Morse_theory" rel="nofollow">Circle-valued Morse theory</a> and <a href="http://www.pnas.org/content/108/20/8122.full" rel="nofollow">Morse 2-functions</a> deal with Morse functions to S<sup>1</sup> and to R<sup>2</sup> correspondingly, and are quite useful.<br><br> And so, in order to feel I have a bit more of a grip on the meta-mathematical conceptual framework of the Morse Lemmas, I'm very much interested in the following question:</p> <blockquote> Let $f\colon\, M\to N$ be a smooth function, with non-degenerate critical points (whatever that means in context). What conditions do I have to impose on M and on N to obtain reasonable analogues of the Morse Lemmas? </blockquote> <p>By reasonable analogues, I mean lemmas which explicitly relate $f^{-1}(N^{\prime})$ to $f^{-1}(N^{\prime\prime})$ up to diffeomorphism (by some sort of generalized handle attachment operation?), where $N^{\prime}\subseteq N^{\prime\prime}\subseteq N$ are some reasonable analogues in N to $(-\infty,-\epsilon]$ and $(-\infty,\epsilon]$ correspondingly.<br><br> Is there any work, or any results along these lines? Is this all well-known (and easy?), is it open, or is it known to be impossible (as in: "if the target isn't R then something goes disastrously wrong")?</p> http://mathoverflow.net/questions/80213/most-general-context-for-the-morse-lemmas/86955#86955 Answer by Liviu Nicolaescu for Most general context for the Morse Lemmas Liviu Nicolaescu 2012-01-29T12:43:49Z 2012-01-29T12:43:49Z <p>The questions that you asked are addressed by the once very sexy field of catastrophe theory. The story is a bit too long to tell here. The conditions you are asking for are called stability conditions. Hassler Whitney is one of the pioneers. He gave beautiful answers for $f: M\to N$, $\dim N=\dim M=2$. (The folds and cusps were discovered by him.) See the two volume book of Arnold-Varchenko-Husein-Zade, or the Golubitsky-Guillemin book: <em>Stable Mappings and Their Singularities</em></p>