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Let manifold $M$, and let a Morse function $f:M \to \mathbb{R}$ the answer to my question follows from Morse theory. For fixed $f$ the manifold then decomposes as for example here.

Now, what happens if I am forced to use smooth function to $\mathbb{R}^n$, namely I consider $$\widetilde{f}:M \to \mathbb{R}^n$$

I am not an expert in topology (to say the least) but am interested in the link between the image of a given $\widetilde{f}$ and topology of $M$.

Any comments \ references are welcome!

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    Then each component of your function might be a morse function, so you have $n$ different morse functions to work with, generically. – 2017-01-02
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    @John Hughes is there more to say about this? Can it be viewed as a surgeon theory, where perhaps $\widetilde{f}$ is how I glue $\text{im}(\widetilde{f})$ to M? – 2017-01-02

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A Morse function $f : M \to \mathbb R$ has an image we understand well (an interval if $M$ is connected). On the opposite, consider $M$ embedded in $\mathbb R^n$ and $i : M \to \mathbb R^n$ the inclusion : we have no new informations about the topology of $M$ as general functions don't have "simple models", e.g no higher Morse lemma and so one.

A small remark : for complex algebraic varieties there is also a complex Morse theory, where "Morse functions" are called Lefschetz fibrations, which are functions $f : X \to \mathbb C$ with non-degenerate critical points, and the theory is really nice.

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    Thanks for your post, but let me insist a bit further, what happens if we start with a complicated manifold (say, Betti numbers are large) and we map this space onto a contractible space, maybe $[0,1]^n$. Why isn't it fair to say that we map complex manifold into "an image we understand"? – 2017-01-06
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    What do you know about Morse theory @them ? – 2017-01-06
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    I have skimmed over the first part of M. Goresky and R. MacPherson, Stratified Morse theory, and few other papers on the subject, and over the years I have seen some use of Morse theory mostly in connection with the Euler characteristic of excursion sets of random fields. I have a general familiarity with topology (set topology, differential geometry), but all this is not really entrenched in my mathematical toolkit. – 2017-01-06
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    Ok. Because basically I think lot of lemmas become wrong. The size of the fibers will become smaller but your have more freedom on the base space and non "canonical way" of moving. That's why I gave example of Picard-Lefschetz theory : I suspect there is no reasonable generalizations of Morse theory. Now I'm not an expert, so maybe other people will tell you other viewpoint on the subject. – 2017-01-06