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

  1. 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 $f^{-1}(-\infty,\epsilon]$.
  2. 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 fi).
    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")?

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Could you clarify what you mean by non-degenerate critical points? I don't understand the phrase "critical points which don't vanish after a small perturbation". –  John Pardon Nov 6 '11 at 18:58
    
When M and N are smooth manifolds, I mean "non-degenerate critical points" in the usual sense- that (in local coordinates) the Hessian matrix is nonsingular. Where M and N are not manifolds, I think that I mean "an isolated critical point p of f is nondegenerate is there exists an open neighbourhood U of f in the relevant topology on $C^\infty(M,N)$ such that any g in U has exactly one critical point in a neighbourhood of p", and the analogous notion for critical submanifolds (which I obviously care about less). –  Daniel Moskovich Nov 6 '11 at 20:22
    
I editted the question (and added an expository paragraph). Does this improve things? –  Daniel Moskovich Nov 6 '11 at 20:30
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I think the difficulty is there's so many ways to relax conditions, and not all of them share a common outlook so it's hard to expect a uniform yet greatly more general set-up. Morse theory on stratified spaces has been studied in quite a bit of detail. Goresky and Macpherson's book is all about this. In particular it includes manifolds with boundary and corners. I've used these techniques to find nice "Poincare dual" decompositions of the 4-manifolds with corners that come up in the interpretation of the Lawrence-Krammer representation. –  Ryan Budney Nov 6 '11 at 20:50
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There are theorems to the effect that if we suitably count the singular circles mod 2 of a generic function from an odd dimensional closed manifold to the plane (such functions have circles of fold points and isolated cusps appearing on the folds), then the resulting number coincides with the Euler-Kervaire semi-characteristic (= the sum of the ranks of mod 2 homology groups up to one-half the dimension). This suggests that there ought to be an analog of the Morse Lemma in this situation. –  John Klein Nov 6 '11 at 21:23
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1 Answer

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

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Thanks! I'll retag, and go to the library to borrow these references. –  Daniel Moskovich Feb 2 '12 at 0:00
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