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!
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.