Decomposing proper map into closed embedding and proper submersion

Question

Suppose that $f: X \to Y$ is a smooth proper map between two smooth manifolds. Is it always possible to represent $f$ as a composition of a closed embedding $g: X \to Z$ with a proper submersion $h: Z \to Y$?

Motivation:

Firstly, closed embeddings and proper submersions are in a sense the simplest kinds of proper maps, so in some cases we could hope reduce proofs about proper maps to those two cases.

Secondly, projective morphisms in algebraic geometry are defined precisely as morphisms that admit such decomposition for $Z = \mathbb{P}^n_Y$, and so far all examples of proper maps that I have come up with are in a sense similar.

Finally, if $X$ is compact, we always have such decomposition as $X \to X \times Y \to Y$, where the first map is the closed embedding of the graph of $f$ and the second map is the projection onto $Y$.

Answer 1

Yes. Choose a smooth (but not necessarily closed) embedding $i:X\to W$ where the manifold $W$ is compact, for example a sphere. Together $i$ and $f$ give a smooth map $X\to W\times Y$ that is both proper and an embedding.