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Pictures in introductory texts to Morse theory are often drawn as to interpret a Morse function as a height function. Typically, an embedding of a torus into $\mathbb{R}^3$ is drawn, and the Morse function is then the height function by projecting onto one component (call the projection $\pi$).

This is a great picture because I have the feeling that any embedding can be perturbed to give a Morse function that way. Such a construction is used - as far as I can tell - in Lurie's definition of $(\infty,1)$-categories of bordisms. But is this a good picture for any Morse function?

In formulas, is every Morse function $f: M \to \mathbb{R}$ of following the form: $$f: M \hookrightarrow \mathbb{R}^n \stackrel{\pi}{\to} \mathbb{R}$$

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I wonder, if the question were stripped of infinity categories, Lurie's theory of bordisms and Morse theory, would it be allowed to remain on mathoverflow? –  Daniel Pomerleano Aug 27 at 14:15
    
@Daniel, since the answer is so easy (embarrassingly for me), I should have asked it on math.stackexchange.com. But sometimes I can't tell how hard the problem is in advance. –  Turion Aug 27 at 14:20

2 Answers 2

up vote 7 down vote accepted

The formal answer is yes. Moreover, the function does not have to be Morse, just any smooth function.

Indeed let $f\colon M\to \mathbb{R}$ be any smooth function. Let us fix an imbedding $i\colon M\to \mathbb{R}^{n-1}$; for large $n$ it always exists. Consider the imbedding $(i\times f)\colon M\to \mathbb{R}^{n-1}\times \mathbb{R}=\mathbb{R}^n$. This is the required imbedding, and projection to the last coordinate function is the hight function whose restriction to $M$ equals $f$.

UPDATE Just noticed that Peter Michor posted the same answer few seconds earlier.

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This is trivially true: take an embedding $g:M\to \mathbb R^{n-1}$ and consider $(g,f):M\to \mathbb R^{n-1}\times \mathbb R\to \mathbb R$.

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Just noted that semyon alesker posted the same answer a few seconds earlier. –  Peter Michor Aug 27 at 12:24

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