Under which conditions does a m-manifold $M^m$ admit a deformation retraction to a
small neighborhood of some k-dimensional subpolyhedron?
Or, under which conditions is the identity map $id_M$ of a smooth m-manifold $M^m$ isotopic through embeddings to an embedding into a small neighborhood of some k-dimensional subpolyhedron? Here we require that the isotopy is the identity on the subpolyhedron.

Probably(?) the latter holds, if the manifold admits a Morse function
with critical points whose Morse indices are all at most k.
In the case of k=m-1, this result is e.g. used in the proof of h-principles for Diff-invariant partial differential relations on open manifolds, compare [Gromov, Partial differential relations, p.37].

Where can one find a proof of this or related results in the literature?
Or what are the key ingredients to provide such a proof?

In the case of an open manifold and k=m-1,
Gromov indicates [loc.cit, p.37] that one can use the Morse complex associated to a Morse function without maxima to obtain a suitable subpolyhedron. Which 'Morse complex' is meant here, and how does one obtain from it a suitable subpolyhedron of M? On the other hand, in Exercise (b), it is suggested that one constructs a subpolyhedron by using that M embeds into to the complement of the barycenters of the m-simplices of a triangulation.
I am also aware of [Eliashberg and Mishachev, Introduction to the h-Principle, p.38], where it is suggested, that one can pick a triangulation of M and choose disjoint paths from the barycenters to infinity. Then an embedding into a neighborhood of the m-1 skeleton is obtained by using these 'paths to infinity' to construct a suitable isotopy. It is however unclear to me how one makes this last step precise.

Thanks!