Most general context for the Morse Lemmas

2011-11-06T14:25:37Z

Among the foundational results in differential topology are the Morse lemmas:

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 $f^{-1}(-\infty,\epsilon]$.
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_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.

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".

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?).

On the target side, what about if we want the target to be something other than $\mathbb{R}$? Circle-valued Morse theory and Morse 2-functions deal with Morse functions to S¹ and to R² correspondingly, and are quite useful.

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:

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?

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.

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")?

Answer by Liviu Nicolaescu for Most general context for the Morse Lemmas

2012-01-29T12:43:49Z

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: Stable Mappings and Their Singularities

Most general context for the Morse Lemmas - MathOverflow

Most general context for the Morse Lemmas

Answer by Liviu Nicolaescu for Most general context for the Morse Lemmas